Limit of the integrals of a sequence of function.

Question

Sorry, maybe it's a dumb question.

Let $f_n$ be a sequence of bounded functions such that $f_n\to f$ pointwisely. Is that true then that $\lim \int f_n d\mu=\liminf \int f_n d\mu=\limsup \int f_n d\mu$?

Answer 1

Let $f_n= I_{(n, n+1)}$ for $n$ even and $f_n=0$ for $n$ odd. Then $f_n$'s are bounded, $f_n\to 0$ pointwise but $\int f_n$ (with respect to Lebesgue measure) oscillates between $0$ and $1$.