Let X be a set and A a σ-algebra of subsets of X. (a) What does...

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%] (b) If f and g are measurable and α, β ∈ R show that the function αf + βg is also measurable. [7%] (c) (i) Suppose that f is a measurable function. Is |f| measurable? (Give a proof or a counterexample.) [3%] (ii) Suppose that |f| is a measurable function. Is f measurable? (Give a proof or a counterexample.) [3%] (iii) Let X = R and let f(x) = 3 if x is rational and f(x) = 1 if x is not. What is the smallest σ-algebra of subsets of R with respect to which f is measurable?

Expert Solution

Colby Messinger answered 2 years ago

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...

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Discrete Math Course. On Z, let B be the set of subsets A of Z where...

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Let x be a set and let R be a relation on x such x is...

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