# Uniform convergence of averages

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 $g(T) := \frac{1}{2T} \int_{-T}^{+T} |f(x)|^2 \, dx$?

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