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I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong:

Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with $\displaystyle\forall x \in E\setminus N: \lim_{k \to \infty}f_k(x)=f(x)$, where $\lambda(N)=0$, $E,N \subset \mathbb{R}^n$ and $\displaystyle\lim_{k \rightarrow \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$.

Is it always true that $$\lim_{k \rightarrow \infty}\int_E |f-f_k(x)|d\lambda=0 ?$$

I see the striking similarity to Lebesgue's dominated convergence theorem, if one could use $\displaystyle\lim_{k \to \infty}\int_E f_k(x) d\lambda=\int_E f(x) d\lambda$ to find some majorant $g$ for our $f_k$ it would be true, especially it would be true when the $f_k$ converge against $f$ from below.

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Or, on the other hand, to find a counterexample you should look for a sequence with no integrable majorant. – GEdgar Dec 6 '11 at 18:49
Yes I tried that, however the problem is that the integrals still have to converge... and all $f_k$ have to be integrable. – Listing Dec 6 '11 at 18:58
Keep trying..... – GEdgar Dec 6 '11 at 19:21
up vote 4 down vote accepted

We have $f_k$ converges to $f$ pointwise almost everywhere, $f_k$ and $f$ are non-negative, and $\int f_k \rightarrow \int f $.

Note that, since the $f_k$ and $f$ are nonnegative: $$| f_k-f\,|\le f_k + f \quad \Rightarrow\quad f_k + f -|f_k-f\,|\ge 0.$$

By Fatou's Lemma: $$ \tag{1}\liminf_{k\to \infty} \int ( f_k + f -|f_k-f\,|\,)\ \ge \int \liminf_{k\to \infty} \,(f_k + f -|f_k-f\,|)\,. $$

Since $\int f_k\rightarrow\int f$, we have $$ \tag{2}\liminf_{k\to \infty} \int ( f_k + f -|f_k-f\,|\,)= 2\int f -\limsup_{k\to \infty}\int|f_k-f\,|. $$

Since $f_k\rightarrow f$ almost everywhere, we have $$ \tag{3}\int \liminf_{k\to \infty} \, (f_k + f -|f_k-f\,|\,)= \int 2 f. $$

Substituting the expressions on the right hand sides of (2) and (3) into (1) gives: $$ 2\int f -\limsup_{k\to \infty}\int|f_k-f\thinspace|\ge\int 2 f\ ; $$whence $$ \limsup_{k\to \infty}\int|f_k-f|\le0. $$

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Let $g_k^+:=(f_k-f)_+$ and $g_k^-:=(f-f_k)_+$, so $|f-f_k|=g_k^+ + g_k^-$ and $f_k-f=g_k^+ - g_k^-$. First note that $g_k^- \le f$ since $f$ is nonnegative, and $g_k^- \to 0$ pointwise almost everywhere. By dominated convergence, $\int g_k^-\to 0$. But $$\int (g_k^+ + g_k^-) = \int(f_k-f+2g_k^-) = \int f_k - \int f + 2\int g_k^- \to 0$$

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