Lebesgue integrable

[[concept]]

Topics

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Lebesgue Integrable

Let $E\subset \mathbb{R}$ be measurable. A measurable function $f:E\to \mathbb{R}$ is Lebesgue integrable over $E$ if ${\int }_E|f|<\infty$

NOTE

Recall that if $f$ is measurable, we have $f=f^{+}-f^{-}$ Where $f^{+},f^{-}$ are both nonnegative measurable functions. Then $|f|=f^{+}+f^{-}$ is measurable too, since sums and products of measurable functions are measurable. We can define

${\int }_E|f|={\int }_E{f}^{+}+{\int }_E{f}^{-}$

Note that LHS is finite if and only if one of the terms on the RHS is infinite. This means that in order for $f$ to be integrable, both $f^{+}$ and $f^{-}$ need to be integrable.

References

References

See Also

Mentions

Mentions

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