Question

(1) Let (, A, i) be a measure space. {AnE: Ae A} is a o-algebra of E, contained in (a) Fix E E A. Prove that Ap = A. (b) Let uE be the restriction of u to AĘ. Prove that iE is a measure on Ag. (c) Suppose that f : Q -» R* is measurable (with respect to A). Let g = the restriction of f to E. Prove that g : E ->R* is measurable (with respect to AE) fE be (d) Suppose that f is integrable on E. Prove that g duE AnE AnE for any A E A. (You must show that f and g are integrable on AnE.

Answer 1

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(A)  Fix E E A. Prove that Ap = {AnE: A E A} is a a-algebra of E, contained in A.

SOL::-

Recall the definition of a $\sigma-$algebra, $\mathcal A$ is said to be an $\sigma-$algebra of $\Omega$, if

i)
EA

ii) $\mathcal A$ is closed under complement.

iii) $\mathcal A$ is closed under countable union.

Note that i)
E E A=> E E A

ii) Let
ВЕАВ Е
, then to show
BC E A
. Note that by def there exists
A E A
, such that $B=E\cap A$. Now since $\mathcal A$ is an sigma algebra, implies
E, AE A C
, now
Bc E B E-(An E)= En (ANE) = En (Ac U E°) E A.
as
(Ac UE) E A
.

iii) Take any countable family , then for all i, there exists
A, E A
, be such that
Bi AnEUB En (JA) e AB iEN iEN

Hence $\mathcal A_E$ is a $\sigma-$algebra of E.

B)  Let be the restriction of u to AE. Prove that E is a measure on AE

SOL::-

Recall $\mu$ is said to be a measure on a $\sigma-$algebra $\mathcal A$, if

i) $\mu(A)\ge 0$, for all
A E A
, ii) $\mu(\phi)=0$, iii) $\mu(\bigcup_{i\in \mathbb Z} A_i)= \sum_{i \in \mathbb Z}}\mu(A_i)$ , for pairwise disjoint A_i's

Note that $A\in \mathcal A_E=> A\in \mathcal A=> \mu_E(A)=\mu(A)\ge 0$

Also note that

and $\mu_E(\bigcup_i A_i)=\mu(\bigcup_i A_i)=\sum_i\mu(A_i)=\sum_i \mu_E(A_i)$ , hence a measure.  

C) Suppose that f -> R* is measurable (with respect to A). Let g = f|e be the restriction of f to E. Prove that g: E -> R is measurable (with respect to AE

SOL::-

Note that it is enough to show that inverse image of open set $U\subset \mathbb R$ , is in the sigma algebra.

Note that is measurable implies $f^{-1}(U)\in \mathcal A$ , then
ΤU)- U uEEAΕ -1 (U) Λ ΕΕ A. g
, hence measurable.

D) Suppose that f is integrable on E. Prove that f du g duE AnE AnE AnE for any A CA. (You must show that f and g are integrable on A intersection E

d) Note that since f is integrable there exists simple functions $\{f_i\}$ such that $\int_Ef_id\mu\to \int_Efd\mu$ uniformely.

Now for simple function $\int_{E\cap A}f_id\mu\le \int_{E\cap A}f_id\mu$ , gives us thus $\int _{E\cap A} f d\mu =\lim_{i\to \infty }\int_{E\cap A}f_id\mu<\infty$ exists and hence f is integrable on $E\cap A$. And now $f|_{E\cap A}=g$ , gives us g is integrable as so is the function $f|_{E\cap A}$, and integration both side we get

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