Composition of measurable maps is measurable

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TODO: write this - as it's mostly copied from the measurable map page

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Statement

Given two measurable maps their composition is measurable:

• $f:(A,\mathcal{A})\rightarrow(B,\mathcal{B})$ is measurable (same as saying: $f:A\rightarrow B$ is $\mathcal{A}/\mathcal{B}$-measurable) and
• $g:(B,\mathcal{B})\rightarrow(C,\mathcal{C})$ is measurable

then:

• $g\circ f:(A,\mathcal{A})\rightarrow(C,\mathcal{C})$ is measurable.

In effect:

• $\mathcal{A}/\mathcal{B}$-measurable followed by $\mathcal{B}/\mathcal{C}$ measurable $=$ $\mathcal{A}/\mathcal{C}$-measurable

Proof

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TODO: See page 6 if help is needed (it wont be)

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