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## Main Question or Discussion Point

Suppose ##\mu:\mathcal{F}\rightarrow[0,\infty)## be a countable additive measure on a ##\sigma##-algebra ##\mathcal{F}## over a set ##\Omega##. Take any ##E\subseteq \Omega##. Let ##\mathcal{F}_{E}:=\sigma(\mathcal{F}\cup\{E\})##. Then, PROVE there is a countable additive measure ##\nu:\mathcal{F}_{E}\rightarrow [0,\infty)## such that ##\nu(A)=\mu(A)## for any ##A\in\mathcal{F}##. I already know the measure extension theorem. But it is based on algebra and the extension is only about ##\sigma(\mathcal{F})##. Can someone give me hint? Or how can I make use of measure extension theorem.

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