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

marimo
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    I think it is because we define $f$ is Lebesgue integrable if $\int_E f <\infty$. – Lev Bahn Nov 12 '18 at 06:20
  • We do in some cases, e.g., $f$ nonnegative. However, note that $\int_E f=\infty$ "exists", but $f$ is not integrable. – user10354138 Nov 12 '18 at 06:29
  • Why do we restrict the definition of integrability to the case $\int |f| < \infty$? What if we allow $\infty$? – marimo Nov 12 '18 at 06:44
  • We do allow integrals to have the value $\pm\infty$. On the other hand, if certain regularity conditions (such as integrability, $L^p$-condition, uniform integrability, etc) are met, we can say much interesting things, so we often restrict our focus on such functions. – Sangchul Lee Nov 12 '18 at 07:02

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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.

RRL
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