Convergence of Integrands and Integrals

Question

Suppose $E \subset \mathbb{R}$ is compact. Is it possible to find a sequence of positive continuous functions $f_n: E \to \mathbb{R}$ such that for every $x \in E$ we have $$f_n(x) \to f(x)$$ for some $f \in L^\infty(E)$ and $$\int_E \! f_n(x) \, dx \to C < \infty$$ but $$C \neq \int_E \! f(x) \, dx?$$ I would think not, but none of the standard convergence theorems seem to do the trick. All integrals are supposed to be Lebesgue integrals.

Answer 1

Let $f$ be any nonnegative continuous function with support contained in $[0,1]$, and $$f_n(x)=nf(nx).$$ To get a positive function, just add something like $e^{-x^2}$ to each $f_n$.