Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X....

Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X.

(a) What does it mean for a function f : X → R to be measurable? [2%] If f and g are measurable, show that the function f − g is also measurable. [6%]

(b) Let (fn) be a sequence of measurable functions.

(i) What does it mean to say that (fn) converges pointwise to a function f? [2%]

(ii) If (fn) converges pointwise to f, show that f is a measurable function.

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Expert Solution

Q3 [17% ] Let X be a set and A a σ-algebra of subsets of X.

(a) What does it mean for a function f : X → R to be measurable? [2%] If f and g are measurable, show that the function f − g is also measurable. [6%]

(b) Let (fn) be a sequence of measurable functions.

(i) What does it mean to say that (fn) converges pointwise to a function f? [2%]

(ii) If (fn) converges pointwise to f, show that f is a measurable function.

Colby Messinger answered 1 year ago

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