integrable function

The Fourier transform of integrable functions have additional properties that do not always hold.

Since the integrand is an integrable function of t, the integral expression makes sense.

The convolution defines a product on the linear space of integrable functions.

The derivative of any integrable function can be defined as a distribution.

Suppose that is a locally integrable function defined on an open set .

An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere.

It can also be used to show that if an integrable function, and both have compact support then .

The typical example is the free particle with the space of square integrable functions on three dimensional space.

The integral of an integrable function on a set of probability 0 is itself 0.

In particular, any locally integrable function has a distributional derivative.