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:

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

then:

• [ilmath]g\circ f:(A,\mathcal{A})\rightarrow(C,\mathcal{C})[/ilmath] is measurable.

In effect:

• [ilmath]\mathcal{A}/\mathcal{B} [/ilmath]-measurable followed by [ilmath]\mathcal{B}/\mathcal{C} [/ilmath] measurable [ilmath]=[/ilmath] [ilmath]\mathcal{A}/\mathcal{C} [/ilmath]-measurable

Proof

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

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