We define:
$$f: I = [0,1] \rightarrow \mathbb{R}: x \mapsto f(x) = \sin(x)1_{[0,1] - \mathbb{Q}} + x1_{[0,1]\cap\mathbb{Q}} $$
where $1_A$ is the characteristic function on A.
Show that $f$ has a gauge integral on [0,1] and calculate it.
I've come across this problem and I don't have a clue on how to solve it. I thought about using the definition of gauge integral but that would require me to "guess" what the value of $\int_I f$ actually is. Having no clue what $\int_I f$ is worth I've been thinking about using Cauchy's criterion instead but without much success.
Any help or clues appreciated.
Because $f$ is Lebesgue-integrable, then the gauge integral is the same that the Lebesgue integral. Now, for this:
$\int_I f=\int_I sin(x)=-cos(1)+1$
The first $=$ works because the Lebesgue integral over a $0$-measure set is $0$