Why is not $\infty$ allowed as a values of Lebesgue integral?

Question

$\infty$ is allowed as a value of Lebesgue measure $m(E)$ and function $f(x)$, but why do not we say $\int_E f= \infty$?

Answer 1

The Lebesgue integral of a nonnegative function can be $+\infty$. However, we define integrability for nonnegative functions as the condition that $\int_E f < +\infty$.

To extend the integral to general functions we define

$$\int_E f =\int_E f^+ - \int_E f^-$$

This is always possible only for the class of integrable functions where $\int_E f^+ + \int_Ef^-=\int_E |f| < +\infty$ since $\infty - \infty$ is indeterminate.