Lebesgue integration over domains

Question

Is $\int\limits_{X_1\setminus X}f\mathrm=\int\limits_{X_1}f\mathrm -\int\limits_{X}f$ always true on disjoint interval $X_1\setminus X$

I tried to show from addivity over domains of Integration but I could not reach anything. Any help will be appreciated.

Answer 1

If $f$ is integrable then $I_{X_1}=I_{X_1\setminus X}+I_{X}$. Multiply by $f$ and integrate to get $\int_{X_1}f= \int_{X_1\setminus X}f+\int_X f $. Now subtract $\int_X f $ from both sides.

[ $I_A(x)=1$ if $x \in A$ and $0$ if $x \notin A$].