Application of Lebesgue's dominated convergence theorem

Question

I want to apply Lebesgue's dominated convergence theorem to a function $g(x)=\int_{-\infty}^x f(t) dt$ to discuss the situation when $x\to -\infty$, where $x\in\mathbb{R}$ and $f$ is integrable. But in the statement of the theorem, the sequence of measurable functions is countable and $n\to +\infty$. Do I need to set another function $h(x)=\int_{-\infty}^{-x}f(t)dt$ and the characteristic functions to be $\chi_{[-\infty,-n]}(t)f(t)$to apply the theorem? Or I can apply the theorem directly to the original function $g(x)$?