Semester 2 Theorems

Question 1

Tonelli-Fubini for double sums

Answer 1

Consider a double series Em,n Xmn ,with Xmn £ R and assume that for some C £ R, for any N £ Natural numbers: Emn |Xmn| =< C;
Then EmEn Xmn = EnEm Xmn, both converge

Question 2

Dominated Convergence for sums

Answer 2

Consider (Xn),(Yn) in R where we assume: E(Yn) converges absolutely, for almost all n £ Natural numbers, |Xn|=<|Yn|.
Then E Xn , is absolutely convergent.

Question 3

Monotone conv for sums

Answer 3

For m,n £ Natural numbers, assume Xmn £ Rinf, Xmn>=0. Moreover, for any M>=m, assume XMn >= Xmn for any n £ Natural numbers.
Then limm En Xmn = En limm Xmn

Question 4

Fatou’s lemma (sums instead of integrals)

Answer 4

If for any m,n £ Natural numbers, Xmn £ Rinf, Xmn>=0: En supm infM>=m XMn =< supm infM>=m En XMn

Question 5

Fatou’s lemma for sums

Answer 5

If for m,n £ Natural numbers, Xmn £ Rinf and Xmn>=0, then
En limm inf Xmn =< limm inf En Xmn

Question 6

Fatou’s lemma for sums

Answer 6

If for m,n £ Natural numbers, Xmn £ Rinf and Xmn>=0, then
En limm inf Xmn =< limm inf En Xmn

Question 6

Fatou’s lemma for sums

Answer 6

If for m,n £ Natural numbers, Xmn £ Rinf and Xmn>=0, then
En limm inf Xmn =< limm inf En Xmn

Question 7

Levi’s Theorem on monotone convergence

Answer 7

Consider (fm), a monotone increasing sequence of measurable functions fm:B->[0,inf].
int lim fm(x) dL = lim int fm(x) dL =: f(x)

Question 8

Lebesgues Dominated convergence

Answer 8

Let B £ L, fm:B->Rinf for any m £ Natural numbers is measurable such that almost everhwere on B, f(x):= lim fm(x) exists and |fm(x)| =< F(x) with F £ L(B).
Then f £ L(B), fm £ L(B) and
lim int fm(x) dL = int f(x) dL

Question 9

Fatou’s Lemma (integrals)

Answer 9

Let B £ L, assume that for every m £ Natural numbers, fm:B->[0,inf] is measurable.
Then
int lim inf fm(x) dL
=<
lim inf int fm(x) dL

Question 10

Cavalier’s Principle

Answer 10

Assume that A c R^L is lebesgue measurable, L=m+n. Then for almost all y £ R^L, Ay is lebesgue measurable. Moreover Rn->Rinf, y->Lx(Ay) defines a measurable function on Rn,
L(A) = intRn Lx(Ay)dLy

Question 11

Product formula

Answer 11

Let A = A1 x A2 c RL with A1 c Rm, A2 c Rn, L = m+n.
1) If A is measurable, then L(A) = Lx(A1).Ly(A2)
2) if A1 and A2 are measurable then so is A.

Question 12

Fubini-Tonelli

Answer 12

If X and Y are σ-finite measure spaces, and if f is a measurable function, then
intX(intY |f(x,y)| dy)dx
=
intY(intX |f(x,y)| dx)dy.
Besides if any one of these integrals is finite, then you can remove the absolute values.