An application of the Dominated Convergence Theorem

Question

I read the following argument. Consider an integrable function $f$ satisfying that:

$$ F(x) - F(a) = \int_a^x f(t) dt, $$

for $a \leq x \leq b$ and some function $F$. Then

$$ F(x+h) - F(x) = \int_a^b 1_{(x, x+h)}(t)f(t)dt \to 0, $$

as $h \downarrow 0$ by the dominated convergence theorem.

Question: How is DCT used in here, please?

First, I can see that a sequence of functions $f_h(t):=1_{(x, x+h)}(t)f(t) \to 0$ as $h\downarrow 0$ for all $t\in[a,b]$. Then I need to find an integrable function $g$ such that $|f_h|\leq g, a.e.$ I guess that I can simply take $g$ to be $f(t)$. Is my argument correct, please? Thank you!

Answer 1

Yes, your idea is correct. Note that \begin{align*} |f_h(t)|&=\begin{cases} |f(t)|, & t\in[a,b]\cap(x,x+h) \\ 0 , & t\in[a,b]\backslash(x,x+h)\end{cases} \\ &\leq |f(t)| \end{align*} for all $t\in[a,b]$.