Lebesgue Measurable Functions

SUMS, PRODUCTS, AND COMPOSITIONS

1. Definition A function is said to be Lebesgue measurable, or simply measurable, provided that its domain is a measurable subset of and for each real number , the set is measurable.

2. Proposition 1 If is measurable, then for every interval of real numbers, is measurable.

3. Proposition 2 A function is measurable if and only if for each open set , the inverse image of under , , is a measurable set.

4. Proposition 3 If is measurable, then every continuous function is measurable.

5. Proposition 4 If , where and are measurable, then is measurable if and only if its restrictions to and are measurable. In particular, if and , then is measurable if and only is measurable

6. Theorem 5 If and are measurable functions, then for any and (Linearity) is measurable. (Products) is measurable

7. characteristic function:a Lebesgue Measurable Functions

8. Proposition 6 If is a measurable function and is continuous, then the composition is measurable.

9. Proposition 7 For a finite collectionof measurable functions, the functions and ,,also are measurable

10. almost everywhere(a.e) A set ,a property ,if ,p(x) is said to be established almost everywhere(a.e).

SEQUENTIAL POINTWISE LIMITS AND SIMPLE APPROXIMATION

1. Definition：A sequence of functions is said to converge pointwise to the function provided that
2. Theorem 8If is a sequence of measurable functions that converges pointwise almost everywhere to the function , then is measurable
3. Definition A real-valued function is said to be simple provided that it is measurable and takes only a finite number of values.
   • canonical representation of the simple function:on ,where
4. Definition A measurable function is said to be finitely supported provided that it vanishes on the complement of a set of finite measure
5. The Simple Approximation Theorem:if the function is measurable, then there is a sequence of finitely supported, simple functions that converges pointwise on to and has the property that ,on ,for all .().If , then, in addition, is increasing and each ,if is bounded,uniformly converges to

LITTLEWOOD’S THREE PRINCIPLES, EGOROFF’S THEOREM, AND LUSIN’S THEOREM

1. Egoroff’s Theorem Assume that . If is a sequence of measurable functions that converges pointwise on to the function, then for each > 0, there is a closed set for which and uniformly on
2. Lemma 10(a very special case of the Tietze Extension Theorem):If is a closed subset of and is a continuous function, then it has a continuous extension to .
3. Lusin’s Theorem If is a measurable function, then for each , there is a continuous function and a closed subset of for which and on