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Could anyone help me with the question (d) and (e)? I've finished the question (a), (b) and (c).

(a) Let (X,M, μ) be a measure space and T : X → y a mapping from X oni at Y Prove thai (i) N (B C y : T-1 (B) EM} is a σ-alge

(d) Give an example to show that v need not be σ-finite even if μ is. (e) Describe the measure induced by T(x) on (R, B) endo

You don't need to solve the question (a), (b), and (c), and you could use them directly.

And the following 2 are (b) and (c).

(b) Prove that if u is a probability measure, then so is v (c) Show that if u is complete, then so is v

(a) Let (X,M, μ) be a measure space and T : X → y a mapping from X oni at Y Prove thai (i) N (B C y : T-1 (B) EM} is a σ-algebra of subsets of Y. (ii) The set function v on N given by v(B) μ(T-1 (B)) is a measure T(B) is a measure v is called the measure induced by T, or the pushforward of u by T.
(d) Give an example to show that v need not be σ-finite even if μ is. (e) Describe the measure induced by T(x) on (R, B) endowed with R.
(b) Prove that if u is a probability measure, then so is v (c) Show that if u is complete, then so is v
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Could anyone help me with the question (d) and (e)? I've finished the question (a), (b) and (c). You don't need to solve the question (a), (b), and (c), and you could use them directly. And...
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