Scheffé's lemma states that if $f_n$ is a sequence of Lebesgue integrable functions (i.e. $f_n \in L_1$) that converges almost everywhere to another integrable function $f$, then $\int |f_n - f| \, d\mu \to 0$ if and only if $\int | f_n | \, d\mu \to \int | f | \, d\mu$.

The hypothesis that $f$ is integrable is not essential in the 'if' part: since $\int |f_n - f| \, d\mu \to 0$, there exists a sufficiently large $m$ and an $\varepsilon > 0 $ such that $$\int |f_m - f| \, d\mu < \varepsilon < +\infty. $$

Thus, this implies that $f_m - f \in L_1$ and since $f_m \in L_1$ and $L_1$ is a vector space, one has that $f = f_m -(f_m -f) \in L_1$.

Is the hypothesis that $f \in L_1$ essential in the "only if" part?


1 Answer 1


I believe so. Let $f_n= \chi_{[-n,n]}$ on $\mathbb{R}$. Then $f_n \to 1$ pointwise and $\int |f_n| \, d\mu = 2n$ approaches $\int |1| \, d\mu = \infty$ as $n \to \infty$. But we have $\int |f_n -1| \, d\mu = \int 1-\chi_{[-n,n]} \, d\mu=\infty$ for all $n$.


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