If $f, (f_n)$ are Lebesgue integrable, and if $(f_n)$ increasing pointwise to $f$, does it follow that $\int f_n \rightarrow \int f$

Am I correct? $f_n(x)=\chi_{[0,1)}\cdot x^n$ is a counter example

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    $\begingroup$ Isn't that covered by the monotone convergence theorem? (I am rather skeptical about the "increasing-ness" of the sequence you suggest) $\endgroup$ – Clement C. Oct 6 '15 at 21:44

The statement is correct and thus there is no counter example. The fact that $f_n$, $n\in\mathbb N$, and $f$ are integrable is much stronger than the setting in traditional monotone convergence theorem. Thus, we don't need non-negativity.

For each $n\in\mathbb N$, by monotonicity, we have $$ f_1 \le f_n \le f $$ and thus $$ f - f_1 \ge f - f_n \ge 0. $$ Since $f$ and $f_1$ are both integrable, it follows $f-f_1$ is integrable and dominates $f-f_n$. Thus, $f - f_n$ converges to $0$ in $L^1$. That is, $f_n$ converges to $f$ in $L^1$.

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