Dirac measure

A Diracmaß, named after the physicist Paul Dirac, is a special measure in measure theory.

Definition

It is given a measurable space, ie a basic set along with a defined thereon σ - algebra. For each point a corresponding mapping is defined, which assigns the value of each quantity, if it contains, and the value if it does not contain:

The image is then a measure, and is referred to Diracmaß point or point measurement. Because even a probability measure and probability space. When Diracmaß the unit mass is concentrated in point. It follows that the measure is finite, in particular the measure space is σ - finite.

With the help of the characteristic function can be the defining equation by

For all and express.

Dirac integral

Dirac the integral of the function is defined as the integral under the Lebesgue measure Dirac. Instead of the Lebesgue measure the Dirac measure is used to calculate the integral. Thus results for the integral of an arbitrary function f

Grounds

The Figure is a non-negative function of measurable. The Lebesgue integral of the function under the Dirac measure is defined as follows.

Is an arbitrary sequence of simple functions that converge pointwise and monotonically increasing against. A basic function is a non-negative measurable function only takes finite number of functional values ​​. is the number of function values; be the ( measurable ) quantities on which the function takes the value each. The integral of a simple function is thus defined as follows:

If z is not an element of A, then z is certainly not an element of any of the subsets. Then is the Dirac measure of all equal to zero. Consequently, the integral over A is a total zero.

If z element of for any j, then the Dirac measure is equal to 1; the Dirac measure for all other quantities is then equal to zero. For the integral of simple functions, this results in:

Thus, the Dirac integral is equal to the function value at the point Z, where Z is an element of A.

Another proof is carried out as follows:

For all and is

As a one-element subset of the. Inverse images of measurable sets are measurable. Thus, and accordingly, the amount is integrated over the above.

If, as is possible and also an integration.