Paper 1, Section II, I

Probability and Measure | Part II, 2010

State Carathéodory's extension theorem. Define all terms used in the statement.

Let A\mathcal{A} be the ring of finite unions of disjoint bounded intervals of the form

A=i=1m(ai,bi]A=\bigcup_{i=1}^{m}\left(a_{i}, b_{i}\right]

where mZ+m \in \mathbb{Z}^{+}and a1<b1<<am<bma_{1}<b_{1}<\ldots<a_{m}<b_{m}. Consider the set function μ\mu defined on A\mathcal{A} by

μ(A)=i=1m(biai)\mu(A)=\sum_{i=1}^{m}\left(b_{i}-a_{i}\right)

You may assume that μ\mu is additive. Show that for any decreasing sequence (Bn:nN)\left(B_{n}: n \in \mathbb{N}\right) in A\mathcal{A} with empty intersection we have μ(Bn)0\mu\left(B_{n}\right) \rightarrow 0 as nn \rightarrow \infty.

Explain how this fact can be used in conjunction with Carathéodory's extension theorem to prove the existence of Lebesgue measure.

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