. Assume that f ( x, y ) is increasing in x for any fixed y, and decreasing in y for any fixed x. Prove that f ∈ R ( Q ) . 2. (15 pts) Let S ⊂ R

HONORED ADVANCED CALCULUS MID-TERM EXAM

9:10 – 12:40, 4/17, 2012

A COURSE BY CHIN-LUNG WANG

1. (10 pts) Let f be a bounded function on Q = [ a, b ] × [ c, d ] ⊂ _R

. Assume that f ( x, y ) is increasing in x for any fixed y, and decreasing in y for any fixed x. Prove that f ∈ R ( Q ) . 2. (15 pts) Let S ⊂ R

and S

= { y ∈ R

: ( x, y ) ∈ S } .

(a) Prove that S has measure zero if and only if there exists a countable collection of intervals { I

}

with ∑

µ ( I

) < ∞ such that for any x ∈ S, x ∈ I

for infinitely many k’s.

(b) Prove that if S has ( ( m + n ) -dimensional) measure zero, then S

has (n-dimensional) measure zero for almost all x.

3. (15 pts) Let E ⊂ R

be open and F ∈ C

( E, R

) such that F ( 0 ) = 0 and F

( 0 ) is invertible.

(a) Prove that there exists open U 3 0 such that F

( 0 )

F ( x ) = G

◦ G

◦ · · · ◦ G

( x ) on U, where each G

is primitive C

with G

( 0 ) = 0 and G

( 0 ) invertible.

(b) State and prove the change of variable formula for f ∈ C ( _R

) with compact support.

4. (10 pts) Let ω ∈ Ω

( E ) , where E is an open set in R

. Suppose that R

γ

ω = 0 for all C

closed curves γ in E, show that ω is exact on E.

5. (15 pts) Let ¯c ( S ) and c ( S ) be the outer/inner Jordan content for S ⊂ R

.

(a) Show that S is Jordan measurable (i.e. c ( S ) : = ¯c ( S ) = c ( S ) ) if and only if c ( ∂S ) = 0.

(Hint: Show that ¯c ( S ) − c ( S ) = ¯c ( ∂S ) .)

(b) Show that the Jordan content is finitely additive but not σ-additive.

6. (15 pts) Let µ be a non-negative, additive, finite and regular set function on the collection of elementary sets E = { A ⊂ _R

: A = ^S

I

, where I

’s are bounded intervals. } .

(a) Define the outer measure µ

for all subsets of R

and construct the collection of finitely µ-measurable sets M

( µ ) .

(b) Construct the collection of measurable sets M ( µ ) and show that it is a σ-algebra on which µ

is σ-additive.

7. (10 pts) Show that C [ a, b ] is dense in L

[ a, b ] for any p > 0.

8. (a) (5 pts) Consider F = { f

}

⊂ L

([− π, π ]) where f

( x ) = sin kx. Show that F is a closed and bounded subset, but not compact.

(b) (5 pts) Let n

∈ N, k = 1, 2, · · · be a strictly increasing sequence. Show that the set E = { x ∈ [− π, π ] | lim

sin n

x converges } has m ( E ) = 0.

9. (Bonus: 10 pts) Prove the change of variable formula for f ∈ L ( g ( T )) where T ⊂ _R

is open, g : T → _R

is C

, one to one, and det g

( t ) 6= 0 for all t ∈ T. (Hint: Use 3. (b) or Fubini.)

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