Mean Value Theorem for measurable functions

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Suppose that $f$ is a Lebesgue measurable function, $\Omega$ is an open bounded domain in $R^{N}$ and $$ \int_{\Omega}f(x)dx=|\Omega|m,\qquad m\ \ is\ \ a\ \ constant. $$ Under which condition(s) on $f$ we can say $f\equiv m$ ?