What does a Convergent Sequence of Improper Integrals Tell You About the Convergence of the Integrands?

Question

Suppose that $F_{N}(x)$ is a sequence of continous functions defined on $[0, \infty)$ with the property that $\int_{0}^{\infty} F_{N}(x) \cdot dx < \infty$ for all $N$ and further that $\lim_{N \rightarrow \infty} \int_{0}^{\infty} F_{N}(x) \cdot dx = L < \infty$. What can be said about $\lim_{N \rightarrow \infty} F_{N}(x)$? Must $\lim_{N \rightarrow \infty} F_{N}(x)$ converge (at least pointwise) to a function, $F(x)$ on $[0, \infty)$? Can we make a stronger conclusion that $\lim_{N \rightarrow \infty} F_{N}(x)$ must converge uniformly to a continuous function on, $F(x)$ on $[0, \infty)$? Must this function $F(x)$ also be integrable over $[0, \infty)$, i.e. in $L^{1}([0, \infty))$?