Is there a continuous function $f:[0,\infty)\longrightarrow\mathbb R$ such that $\lim_{t\to\infty}\int_{[0,t]}fd\mu$ exists but...?
Let $f(x)=\frac {\sin x} x$ if $x >0$ and $f(0)=1$. Then $\lim_{t \to \infty} \int_{[0,t]}f(x)dx=\frac {\pi} 2$. But $\int_{[0,\infty)} f(x)dx$ does not exist in Lebesgue sense. It is a general fact in Measure Theory that if $\int fd\mu$ is finite then $\in |f|d\mu <\infty$. In this example it is known that $\int |f(x)|dx=\infty$. Hence, it follows that neither $f^{+}$ nor $f^{-}$ is not integrable and $\int fd\mu$ is not defined.