| 1. | Lebesgue's theory defines integrals for a class of functions called measurable functions.
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| 2. | As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable.
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| 3. | Integration theory defines integrability and integrals of measurable functions on a measure space.
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| 4. | The only information available is the basic properties of ?-algebras and measurable functions.
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| 5. | Extending to measurable functions is achieved by applying Riesz-Markov, as above.
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| 6. | It can be used to prove H�lder's inequality for measurable functions
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| 7. | Monotone convergence theorem : Suppose is a sequence of non-negative measurable functions such that
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| 8. | In general, the supremum of any countable family of measurable functions is also measurable.
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| 9. | This shows that Tonelli's theorem can fail for non-measurable functions.
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| 10. | These spaces are spaces of measurable functions on when, and of tempered distributions on when.
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