Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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|improve this question
add comment

1 Answer

I don't think so.

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

share|improve this answer
add comment

Your Answer

 
discard

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.