LET f is Lebesgue Integrable on A where m(A) is finite, Prove that f must be finite almost everywhere on A.
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$\displaystyle \left\{f=\pm\infty\right\}=\bigcap_{n\geq 1}\left\{|f|\geq n\right\}$ and $\displaystyle m(\left\{|f|\geq n\right\})\leq \frac 1n\int_A|f|dm$. Since $\displaystyle m(A)<\infty$, $\displaystyle m(\left\{f=\pm\infty\right\})=\lim_{n\to\infty}m( \left\{|f|\geq n\right\})$ and you can conclude.
And note that the result is true for a $\displaystyle \sigma$-finite measured space $\displaystyle (X,\mathcal A,\mu)$, i.e. a space such that we can find a countable partition of $\displaystyle X$ into sets of finite measure (for example $\displaystyle \mathbb R$ with Lebesgue measure).