Corollary 3

For every integrable function f and every measurable set A, the function $f \varphi_A$ is integrable.

For, it follows from Th. 5 and No. 3, Cor. 5 of Th. 1 that $f \varphi_A$ is measurable, and $N_1(f \varphi_A) \leq N_1(f)$.

We write $\int_A f d\mu = \int f \varphi_A d\mu$ (or $\int_A f \mu$) for every integrable function $f$ and every measurable set $A$. We also write $\int^* f d|\mu|$ (or $\int^*_A f |\mu|$) in place of $\int^* f \varphi_A d|\mu|$ for every numerical function (finite or not) $f \geqslant 0$ (on setting $f(x)\varphi_A(x) = 0$ if $f(x) = +\infty$ and $\varphi_A(x) = 0$).