Not sure where to post about measure theory. None of the forums seems quite right.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose that ##(X,\Sigma,\mu)## is a measure space. A sequence ##\langle f_n\rangle_{n=1}^\infty## of almost everywhere real-valued measurable functions on X is said to converge in measure to a measurable function f, if for all ε>0, ##\mu(\{x\in X:|f_n(x)-f(x)|\geq\varepsilon\})\to 0##.

The book says that it's easily seen that if ##\langle f_n\rangle## converges in measure to two measurable functions f and g, then f=g almost everywhere. I don't see it.

I understand that I need to prove that ##\mu(\{x\in X: |f(x)-g(x)|>0\})=0##, but I don't even see how to begin.

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# Convergence in measure

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