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