Lebesgue Measurable Functions

Simple Functions and Measurable Functions

Characteristic Function

A characteristic function of a set is a function defined by

Simple Function, Step Function

A simple function is a function that can be expressed as a finite linear combination of characteristic functions of measurable sets with finite measure: where are constants. In particular, if each is a rectangle, then is called a step function.

Lebesgue Measurable Function

A function defined on a measurable subset of is Lebesgue measurable, if the set is Lebesgue measurable for all Borel set . To simplify our notation, we shall often denote the set simply by whenever no confusion is possible.

Proposition

The followings are equivalent:

1. is Lebesgue measurable.
2. or or or is measurable for all .

Proposition

The followings are equivalent for a finite-valued function :

1. is measurable.
2. is measurable for all .
3. is measurable for all open sets .
4. is measurable for all closed sets .

Proposition

If is continuous on , then is measurable. If is measurable and finite-valued, and is continuous, then is measurable.

Proposition

Suppose is a sequence of measurable functions. Then are measurable. In particular, if exists a.e., it is measurable.

Proof Note that , and , and other cases are similar.

Proposition

If and are measurable, then

1. The integer powers , are measurable.
2. and are measurable if both and are finite-valued.

Almost Everywhere

We say that a property holds almost everywhere (a.e.) if the set of points where the property fails is a null set.

e.g. We say functions a.e. if has (outer) measure zero.

Proposition

Suppose is measurable, and a.e. . Then is measurable.

Approximation by Simple Functions

Theorem

Suppose is a non-negative measurable function on . Then there exists an increasing sequence of non-negative simple functions that converges pointwise to , namely,

Corollary

Suppose is measurable on . Then there exists a sequence of simple functions that satisfies
In particular, we have for all and .

We may now go one step further, and approximate by step functions. Here, in general, the convergence may hold only almost everywhere.

Theorem

Suppose is measurable on . Then there exists a sequence of step functions that converges pointwise to for almost every .