Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $\{ f_n \}_{n > 0} : \mathbb{R} \to \mathbb{C}$ be a sequence of functions that is uniformly bounded and converges pointwise almost everywhere to the bounded function $f : \mathbb{R} \to \mathbb{C}$.

May I conclude that the sequence of functions defined by $g_n(T) := \frac{1}{2T} \int_{-T}^{+T} |f_n(x)|^2 \, dx$ ($T > 0$) converges uniformly to $g(T) := \frac{1}{2T} \int_{-T}^{+T} |f(x)|^2 \, dx$?

Thank you in advance.

share|cite|improve this question

I don't think so.

Try this example: $f_n (x) = \sin \frac{x}{n}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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