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Suppose $f_{n}:[0,1]\to [0,\infty)$ are nonnegative functions such that $\lim_{n\to \infty}f_{n}(x)=0$ a.e $x\in [0,1]$ and $$\sup_{n}\int\limits_0^1\varphi(f_{n}(x))dx\leq1$$ for some continuous function $\varphi:[0,\infty)\to [0,\infty)$ such that $\lim_{t\to \infty}\frac{\varphi(t)}{t}=\infty$. prove that $\lim_{n\to \infty}\int\limits_0^1f_{n}(t)dt=0$

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Are you sure that $\varphi$ is continuous? Should it read "convex" instead...? – saz Feb 11 '13 at 9:19
The function $\varphi$ is continuous. – Alexander Osorio Feb 11 '13 at 20:30

Fix $\varepsilon>0$. We can find $A$ such that $\frac{\varphi(t)}t>\frac 1{\varepsilon}$ whenever $t>A$. This gives that for each $n$, $$\int_{\{f_n>A\}}\varphi(f_n(x))dx\leqslant 1,$$ hence $$\int_{\{f_n>A\}}f_n(x)dx\leqslant \varepsilon.$$ Use dominated convergence theorem to threat the other part ($|f_n|\leqslant A$).

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