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Fix a measure space $(X,\mathcal{M},\mu)$. Suppose $\{f_n\} \subset L^1$ and $f_n \to f$ uniformly. Then I want to prove the following statement:

If $\mu(X)<\infty$ then $f \in L^1(\mu)$ and $\int f_n \to \int f$.

In some lecture notes, the attempted solution is as follows (I rewrite it in my language):

Since $f_n \to f$ uniformly then there exists $N$ such that $\left\lvert f_n(x) - f(x)\right\rvert < 1$ for all $n \geq N$ and all $x \in X$. Therefore, $$\int \left\lvert f \right\rvert \leq \int \left\lvert f - f_N \right\rvert + \int \left\lvert f_N \right\rvert \leq \int 1+ \int \left\lvert f_N \right\rvert = \mu(X) + \int \left\lvert f_N \right\rvert < \infty$$

so $f \in L^1$.

Then main point that I do not understand here is that, how can we say that $\mu(X) + \int \left\lvert f_N \right\rvert < \infty$? $\mu(X) < \infty$ by definition, however I couldn't see why $\int \left\lvert f_N \right\rvert < \infty$. Is this solution makes sense? If so, can anyone explain this? Thanks!

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$f_N$ is integrable, isn't it? – Davide Giraudo Jan 11 '13 at 21:22
$f_N\in L^1(\mu)$, isn't it? – Ilya Jan 11 '13 at 21:22
@DavideGiraudo: it's even funny, that our comments with 8 seconds difference are equivalent, and still different :) – Ilya Jan 11 '13 at 21:23
@John: since you are done so fast, what if $\mu(X) = \infty$? – Ilya Jan 11 '13 at 21:34
Then I guess these conditions fail. I will try to construct counterexamples. – oeda Jan 11 '13 at 21:54

$$\left|\int_X f_n\,\mathrm d\mu-\int_X f\,\mathrm d\mu\right|\leqslant\int_X|f_n-f|\,\mathrm d\mu\leqslant\mu(X)\cdot\|f-f_n\|_\infty$$

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There is a problem with this proof: one of the stated aims of the proof is to show that f is integrable over X... you have simply assumed this to be true here, as evidenced by the second integral on the left hand side of the first inequality. – Sinister Cutlass Sep 28 '15 at 20:26
Although, I suppose you were simply going off of oeda's legitimate proof for the integrabiity of f. Very well, then. Carry on. – Sinister Cutlass Sep 28 '15 at 20:27
Indeed. $ $ $ $ – Did Sep 28 '15 at 20:29

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