I am having a trouble with proving the following: If $f : (0, a) → \mathbb{R}$ is an integrable function and $$g(x) = \int_x^a \frac{f(t)}{t}dt$$ Then show that $g$ is intergrable and $$\int_0^a g(t)dt= \int_0^a f(t) dt$$
For proving $g$ is integrable I have shown that $1/t$ is continuous in $(0,a)$ and $f$ being integrable $f(t)/t$ is integrable. However I am unable to come with an idea to show the second part, any suggestions? please help.
Hint:
$$\int_0^a \left(\int_x^a \frac{f(t)}{t} \, dt\right) \, dx = \int_0^a \left(\int_0^a \frac{f(t)}{t} \mathbf{1}_{t \geqslant x}\, dt\right) \, dx = \int_0^a\frac{f(t)}{t} \left(\int_0^a \mathbf{1}_{x \leqslant t}\, dx\right) \, dt = \ldots$$
where $ \mathbf{1}_{t \geqslant x} = 1$ if $t \geqslant x$ and $ \mathbf{1}_{t \geqslant x} = 0$ if $t < x$ .