Lebesgue integrable

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Lebesgue Integrable

Let $E \subset R$ be measurable. A measurable function $f : E \to R$ is Lebesgue integrable over $E$ if

\int_{E} | f | < \infty

^def

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

File	Last Modified
Functional Analysis Lecture 10	2025-07-14
Functional Analysis Lecture 11	2025-08-01
Functional Analysis Lecture 12	2025-08-05
Functional Analysis Lecture 13	2025-07-25
Functional Analysis Lecture 9	2025-07-14
Lebesgue Integral	2025-08-01
sums and products of measurable functions are measurable	2025-07-14

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Created 2025-08-01 ֍ Last Modified 2025-09-09

Lebesgue integrable

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