Convergence of an uncountable sequence of functions

Question

I have a function $g(\cdot)$ such that such that $\lim\limits_{t \to \infty}g(t-s)=0$. I need to show that $\lim\limits_{T \to \infty}\int_{-\infty}^tg(T-s)ds=0$, where $0<t\ll T$.
I thought that Lebsegue dominated convergence theorem would work, because the assumption we make implies that $g(T-s)$ can be bounded by some integrable function, for big $T$. However, all convergence theorems I've found deal with countable sequences of functions, while here $T$ belongs to an uncountable set. I would be grateful for suggestions how to deal with this problem. If any additional assumptions need to be made, that's perfectly fine (I'm using this function as a part of my model, so I can play around with it).