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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 algebra, is said to be an algebra of , if
i)
ii) is closed under complement.
iii) is closed under countable union.
Note that i)
ii) Let , then to show . Note that by def there exists , such that . Now since is an sigma algebra, implies , now as .
iii) Take any countable family , then for all i, there exists , be such that
Hence is a algebra of E.
B) Let be the restriction of u to AE. Prove that E is a measure on AE
SOL::-
Recall is said to be a measure on a algebra , if
i) , for all , ii) , iii) , for pairwise disjoint A_i's
Note that
Also note that
and , 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 , is in the sigma algebra.
Note that is measurable implies , then , 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 such that uniformely.
Now for simple function , gives us thus exists and hence f is integrable on . And now , gives us g is integrable as so is the function , and integration both side we get
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