Let $(X,\Sigma,\mu)$ be a measure space, such that $\mu(X)=\infty$.

Let $f_n:X \to \mathbb{R}$ be measurable real-valued functions, which converge uniformly to a function $f$. Suppose that $f_n \in L^1$. Is it true that $f \in L^1$? Does $\int_X f_n \to \int_X f$ hold?

Does the convergence $\int_X f_n \to \int_X f$ hold if we assume in advance that $f \in L^1$?


1 Answer 1


$f$ many not be in $L^1$: consider $f_n(x)=\frac1x\chi_{[1,n]}(x)$, which converges uniformly to $\frac1x\chi_{[1,\infty)}(x)$.

For the last question, consider $f_n(x)=\frac1n\sin^2(x)\chi_{[0,2n\pi]}(x)$ which converges uniformly to $0$.

Added: However, the integrals of a sequence of non-negative functions that converges to a function not in $L^1$ diverge by Fatou's lemma.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.