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let $\{f,f_{n}:n\ge 1\}$ be $\mathbb R$-valued measurable function on $(\Omega,\mathcal A,\mu)$

(a) Assume that $f_{n}\uparrow f$ and that there exist an $(\mathcal A,\mathcal B_{\mathbb R})$ measurable function such that $\int h d\mu< \infty $ and $f_{n}\ge h\quad \forall n$. Then Show that $\int f_{n} d\mu\uparrow \int f d\mu$.

(b) show by counter example that the above hypothesis cannot be dropped.

thanks for help

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(a) Hint: use monotone convergence theorem for the non-negative sequence $(f_n-h)_{n\geqslant 1}$.

(b) Take $f_n(x):=-\frac 1n\chi_{(-n,0)}$: this increases to $0$, but the integral is $1$ for each $n$.

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