Convergence in Sigma-Finite Measure

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Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a $\sigma$-finite measure space.

Let $\left({f_n}\right)_{n \in \N}, f_n: X \to \R$ be a sequence of measurable functions.

Also, let $f, g: X \to \R$ be measurable functions.

Suppose that $f_n$ converges in measure to both $f$ and $g$ (in $\mu$).


Then $f$ and $g$ are equal $\mu$-almost everywhere.


Proof


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