Lebesgue Measurable Functions

Simple Functions and Measurable Functions

As we defined the measurable functions, when the domain of a function is endowed with the Lebsgue -algebra, we call it Lebesgue measurable:

Lebesgue Measurable Function

A function is Lebesgue measurable if it is measurable when the domain is endowed with the Lebesgue -algebra.

Proposition

To simplify our notation, we shall often denote the set simply by whenever no confusion is possible.
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

If and are measurable, then
5. The integer powers , are measurable.
6. 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 .