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Question: Let $(\Omega,\mathcal{F}, \mu)$ be a measure space with $\mu\geq 0$, $f\geq 0$, $\mu(\Omega)<\infty$ and let there be a $A\in \mathcal{F}$ s.t. $\mu(A)=0$ and $f|_{\Omega \setminus A} <\infty$.

Show that $\forall \epsilon>0, \exists$ a bounded measurable function $g$ s.t. $$\mu(\{w\in \Omega: f(w)\neq g(w)\})<\epsilon.$$

It sounds believable because $f$ is finite except for a measure zero set but am not sure where to start. Appreciate a hint.

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$\{x\in \Omega \setminus A: f(x) >N\}$ decreases to the empty set as $N$ increases to $\infty$, so its measure tends to $0$. Take $g=fI_{x: f(x) \leq N}$ where $N$ is so large that $\mu (\{x\in \Omega \setminus A: f(x) >N\}) <\epsilon$.

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