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WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

2009

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## Ch12: Integration on R

n

### 12.1: Jordan Regions

Notations:

(1) R = [a1,b1] × . . . × [an,bn], an n-dimensional rectangle.

(2) A grid G = {R1, . . . ,Rp} on R is a collection of n- dimensional rectangles obtained by subdividing the sides of R.

(3) The volume of R: |R| = (b1− a1) . . . (bn− an).

(3)

Definition (1)

Let E be a given set. The outer sum of E with respect to a grid G on a rectangle R is

V (E ; G) := X

Rj∩E6=∅

|Rj|.

(4)

Remark (12.1)

Let R be an n-dimensional rectangle.

(i) Let E be a subset of R, and let G, H be grids on R. If G is finer than H, then

V (E ; G) ≤ V (E ; H).

(ii) If A and B are subsets of R and A ⊆ B, then V (A; G) ≤ V (B; G).

(5)

Remark (12.1)

Let R be an n-dimensional rectangle.

(i) Let E be a subset of R, and let G, H be grids on R. If G is finer than H, then

V (E ; G) ≤ V (E ; H).

(ii) If A and B are subsets of R and A ⊆ B, then V (A; G) ≤ V (B; G).

(6)

Remark (12.1)

Let R be an n-dimensional rectangle.

(i) Let E be a subset of R, and let G, H be grids on R. If G is finer than H, then

V (E ; G) ≤ V (E ; H).

(ii) If A and B are subsets of R and A ⊆ B, then V (A; G) ≤ V (B; G).

(7)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G.If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not(see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(8)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅,then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(9)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not(see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}.Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

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Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H)= X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(11)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}.Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R|≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(12)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H)= X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R|=V (E ; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(13)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R|≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(14)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R|=V (E ; G).

(ii) If A ⊆ B,then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(15)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254).Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(16)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B,then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G).Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(17)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254).Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative,it follows that V (A; G) ≤ V (B; G).

(18)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G).Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(19)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative,it follows that V (A; G) ≤ V (B; G).

(20)

Proof.

(i) Since G is finer than H, each Q ∈ H is a finite union of Rj’s in G. If Q ∩ E 6= ∅, then some of the Rj’s in Q intersect E and others might not (see Figure 12.3, when the darker lines represent the grid H, the lighter lines represent G\H, and the Rj’s that intersect E are shaded).

Let I1= {R ∈ G : G ∩ E 6= ∅} and I2 = {R ∈ G\I1:R ⊆ Q for some Q ∈ H with Q ∩ E 6= ∅}. Then

V (E ; H) = X

R∈I1

|R| + X

R∈I2

|R| ≥ X

R∈I1

|R| = V (E; G).

(ii) If A ⊆ B, then A ⊆ B (see Exercise 3, p.254). Thus, every rectangle that appears in the sum V (A; G) also appears in the sum V (B; G). Since all |Rj|’s are nonnegative, it follows that V (A; G) ≤ V (B; G).

(21)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R.Since each Rj is nondegenerate,it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p]. Hence V (A; G) = |R| = 1. Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(22)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R.Since each Rj is nondegenerate, it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p].Hence V (A; G) = |R| = 1. Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(23)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R. Since each Rj is nondegenerate,it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p]. Hence V (A; G) = |R| = 1.Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(24)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R. Since each Rj is nondegenerate, it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p].Hence V (A; G) = |R| = 1. Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(25)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R. Since each Rj is nondegenerate, it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p]. Hence V (A; G) = |R| = 1.Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(26)

Example (12.2)

If R = [0, 1] × [0, 1], A = {(x , y ) : x , y ∈Q ∩ [0, 1]}, and B = R\A, then V (A; G) + V (B; G) = 2V (R; G) no matter how fine G is.

Proof.

Let G = {R1, . . . ,Rp} be a grid on R. Since each Rj is nondegenerate, it is clear by the Density of Rationals (Theorem 1.24) that Rj∩ A 6= ∅ for all j ∈ [1, p]. Hence V (A; G) = |R| = 1. Similarly, the Density of the Irrationals (Exercise 3, p.23) implies V (B; G) = |R| = 1.

(27)

Definition (12.3)

Let E be a subset ofRn. Then E is said to be a Jordan region if and only if given ε > 0 there is rectangle R ⊇ E , and a grid G = {R1, . . . ,Rp} on R, such that

V (∂E ; G) := X

Rj∩∂E6=∅

|Rj| < ε.

(The last sum IS the outer sum of ∂E since ∂E = ∂E by Theorem 8.36.)

(28)

Definition (12.4)

Let E be a Jordan region inRn and let R be an n- dimensional rectangle that satisfies E ⊆ R. The volume (or Jordan content) of E is defined by

Vol(E ) := inf

G V (E ; G)

:=inf{V (E ; G) : G ranges over all grids on R}

(29)

Remark (12.5)

If R is an n-dimensional rectangle, then R is a Jordan region inRn and Vol(R) = |R|.

(30)

Proof.

Let ε > 0 and suppose that

R = [a1,b1] × . . . × [an,bn].

Since bj − aj− 2δ → bj − aj as δ → 0, we can choose δ >0 so small that if

Q = [a1+ δ,b1− δ] × . . . × [an+ δ,bn− δ], then |R| − |Q| < ε.

Let G0:= {H1, . . . ,Hq} be the grid on R determined by Pj(G) = {aj,aj+ δ,bj − δ, bj}.

(31)

Proof.

Let ε > 0 and suppose that

R = [a1,b1] × . . . × [an,bn].

Since bj − aj− 2δ → bj − aj as δ → 0,we can choose δ >0 so small that if

Q = [a1+ δ,b1− δ] × . . . × [an+ δ,bn− δ], then |R| − |Q| < ε.

Let G0:= {H1, . . . ,Hq} be the grid on R determined by Pj(G) = {aj,aj+ δ,bj − δ, bj}.

(32)

Proof.

Let ε > 0 and suppose that

R = [a1,b1] × . . . × [an,bn].

Since bj − aj− 2δ → bj − aj as δ → 0, we can choose δ >0 so small that if

Q = [a1+ δ,b1− δ] × . . . × [an+ δ,bn− δ], then |R| − |Q| < ε.

Let G0:= {H1, . . . ,Hq} be the grid on R determined by Pj(G) = {aj,aj+ δ,bj − δ, bj}.

(33)

Proof.

Let ε > 0 and suppose that

R = [a1,b1] × . . . × [an,bn].

Since bj − aj− 2δ → bj − aj as δ → 0, we can choose δ >0 so small that if

Q = [a1+ δ,b1− δ] × . . . × [an+ δ,bn− δ], then |R| − |Q| < ε.

Let G0:= {H1, . . . ,Hq} be the grid on R determined by Pj(G) = {aj,aj+ δ,bj − δ, bj}.

(34)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q.Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(35)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4,let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(36)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R.Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(37)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4,let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G,it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(38)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R.Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(39)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G,it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

(40)

Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

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Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

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Proof.

Then it is clear that an Hj ∈ G intersects ∂R if and only if Hj 6= Q. Hence,

V (∂R; G) := X

Hj∩∂R6=∅

|Hj| = |R| − |Q| < ε.

This proves that R is a Jordan region.

To compute the volume of R by Definition 12.4, let G = {R1, . . . ,Rp} be any grid on R. Since Rj ∩ R 6= ∅ for all Rj ∈ G, it follows from definition that V (R; G) = |R|.

Taking the infimum of this identity over all grids G on R, we find that Vol(R) = |R|.

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Remark (12.6)

Suppose that E is a bounded subset ofRn.

(i) E is a Jordan region of volume zero if and only if there is an absolute constant C, that does not depend on E , such that for each ε > 0 one can find a grid G that satisfies V (E ; G) < Cε.

(ii) E is a Jordan region if and only if Vol(∂E ) = 0.

(iii) If E is a set of volume zero and A ⊆ E , then A is a Jordan region and Vol(A) = 0.

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Remark (12.6)

Suppose that E is a bounded subset ofRn.

(i) E is a Jordan region of volume zero if and only if there is an absolute constant C, that does not depend on E , such that for each ε > 0 one can find a grid G that satisfies V (E ; G) < Cε.

(ii) E is a Jordan region if and only if Vol(∂E ) = 0.

(iii) If E is a set of volume zero and A ⊆ E , then A is a Jordan region and Vol(A) = 0.

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Remark (12.6)

Suppose that E is a bounded subset ofRn.

(i) E is a Jordan region of volume zero if and only if there is an absolute constant C, that does not depend on E , such that for each ε > 0 one can find a grid G that satisfies V (E ; G) < Cε.

(ii) E is a Jordan region if and only if Vol(∂E ) = 0.

(iii) If E is a set of volume zero and A ⊆ E , then A is a Jordan region and Vol(A) = 0.

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Remark (12.6)

Suppose that E is a bounded subset ofRn.

(i) E is a Jordan region of volume zero if and only if there is an absolute constant C, that does not depend on E , such that for each ε > 0 one can find a grid G that satisfies V (E ; G) < Cε.

(ii) E is a Jordan region if and only if Vol(∂E ) = 0.

(iii) If E is a set of volume zero and A ⊆ E , then A is a Jordan region and Vol(A) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i).Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε.Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0.By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε.Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε.Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε.Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0.Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε.Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0.Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region.Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε.Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region.Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Proof.

By Definition 12.3 and 12.4, and Remark 12.1ii, it suffices to prove (i). Let E be a Jordan region of volume zero, and let ε > 0. By the Approximation Property for Infima, there is a grid G such that V (E ; G) < ε. Hence set C = 1.

Conversely, let ε > 0 and suppose that there is a grid G such that V (E ; G) < Cε. Then ∂E = E \Eo ⊂ E implies

0 ≤ α := inf

G V (∂E ; G) ≤ β := V (E ; G) ≤ Cε.

Since ε > 0 was arbitrary, it follows that α = β = 0. Since α =0, we can use the Approximation Property for Infima to choose a grid H such that V (∂E ; H) < ε. Thus E is a Jordan region. Since β = 0, we conclude by Definition 12.4 that Vol(E ) = 0.

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Definition (12.7)

Let E := {E`}`∈N be a collection of subsets ofRn.

(i) E is said to be nonoverlapping if and only if Ej∩ Ek is of volume zero for j 6= k .

(ii) E is said to be pairwise disjoint if and only if Ej ∩ Ek

= ∅for j 6= k .

Notice that since ∅ is of volume zero, every collection of pairwise disjoint sets is nonoverlapping.

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Definition (12.7)

Let E := {E`}`∈N be a collection of subsets ofRn.

(i) E is said to be nonoverlapping if and only if Ej∩ Ek is of volume zero for j 6= k .

(ii) E is said to be pairwise disjoint if and only if Ej ∩ Ek

= ∅for j 6= k .

Notice that since ∅ is of volume zero, every collection of pairwise disjoint sets is nonoverlapping.

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Theorem (12.8)

Let E be a subset ofRn. Then E is a Jordan region of volume zero if and only if for every ε > 0 there is a finite collection of cubes Qk of the same size, i.e., all with sides of length s, such that

E ⊂

q

[

k =1

Qk and

q

X

k =1

|Qk| < ε.

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Corollary (12.9)

If E1 and E2are Jordan regions, then E1∪ E2 is a Jordan region and

Vol(E1∪ E2) ≤Vol(E1) +Vol(E2).

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Corollary (12.10)

Suppose that V is a bounded, open set inRnand that φ :V →Rn is 1-1 and C1on V with ∆φ6= 0.

(i) If E is of volume zero and E ⊂ V , then φ(E ) is of volume zero.

(ii) If {Ek}k ∈Nis a nonoverlapping collection of sets inRn with Ek ⊂ V for all k ∈ N, then {φ(Ek)}k ∈N is a

nonoverlapping collection of sets inRn.

(iii) If E is a Jordan region and E ⊂ V , then φ(E ) is a Jordan region.

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Corollary (12.10)

Suppose that V is a bounded, open set inRnand that φ :V →Rn is 1-1 and C1on V with ∆φ6= 0.

(i) If E is of volume zero and E ⊂ V , then φ(E ) is of volume zero.

(ii) If {Ek}k ∈Nis a nonoverlapping collection of sets inRn with Ek ⊂ V for all k ∈ N, then {φ(Ek)}k ∈N is a

nonoverlapping collection of sets inRn.

(iii) If E is a Jordan region and E ⊂ V , then φ(E ) is a Jordan region.

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Corollary (12.10)

Suppose that V is a bounded, open set inRnand that φ :V →Rn is 1-1 and C1on V with ∆φ6= 0.

(i) If E is of volume zero and E ⊂ V , then φ(E ) is of volume zero.

(ii) If {Ek}k ∈Nis a nonoverlapping collection of sets inRn with Ek ⊂ V for all k ∈ N, then {φ(Ek)}k ∈N is a

nonoverlapping collection of sets inRn.

(iii) If E is a Jordan region and E ⊂ V , then φ(E ) is a Jordan region.

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Corollary (12.10)

Suppose that V is a bounded, open set inRnand that φ :V →Rn is 1-1 and C1on V with ∆φ6= 0.

(i) If E is of volume zero and E ⊂ V , then φ(E ) is of volume zero.

(ii) If {Ek}k ∈Nis a nonoverlapping collection of sets inRn with Ek ⊂ V for all k ∈ N, then {φ(Ek)}k ∈N is a

nonoverlapping collection of sets inRn.

(iii) If E is a Jordan region and E ⊂ V , then φ(E ) is a Jordan region.

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Definition (2)

The inner sum of E with E with respect to G is υ(E ; G) := X

Rj⊂Eo

|Rj|.

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Remark (12.11)

Let R be an n-dimensional rectangle, let E be a subset of R, and let G, H be grids on R. If G is finer than H, then

0 ≤ υ(E ; H) ≤ υ(E ; G) ≤ V (E ; G) ≤ V (E ; H).

This leads us to the following fundamental principle.

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Remark (12.12)

Let R be an n-dimensional rectangle and E be a subset of R. If G and H are grids on R, then

0 ≤ υ(E ; G) ≤ V (E ; H).

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Definition (12.13)

Let E be a bounded subset ofRn and let R be an n- dimensional rectangle that satisfies E ⊆ R. The inner volume of E is defined by

Vol(E ) := sup{υ(E ; G) : G ranges over all grids on R}, and the outer volume of E is defined by

Vol(E ) := inf{V (E ; G) : G ranges over all grids on R}.

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Theorem (12.14)

Let E be a bounded subset ofRn. Then E is a Jordan region if and only if Vol(E ) = Vol(E ).

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## Thank you.

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung

Department of Mathematics National Cheng Kung