Lebesgue measure

Note that here is indicated the Lebesgue measure on the real line.

In this case, the Lebesgue measure of need not be unity.

On the other hand, has a uniform density with respect to and the Lebesgue measure.

The Lebesgue measure also has the property of being σ-finite.

The area of the carpet is zero (in standard Lebesgue measure).

Such distributions are not absolutely continuous with respect to Lebesgue measure.

However these initial data are not generic since they have Lebesgue measure zero.

This is a special case of the preceding example, using Lebesgue measure, but described in elementary terms.

The Lebesgue measure on the real line R is a special case of this.

A similar argument implies that the limit set has Lebesgue measure zero.

The circle group T with the Lebesgue measure is an immediate example.

Then the image has Lebesgue measure 0 in .

Then the standard Lebesgue measure pulls back to under f: .

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.

When discussing sets of real numbers, the Lebesgue measure is assumed unless otherwise stated.

It can be shown that there is no infinite-dimensional analogue of Lebesgue measure.

Often, when working on R, one works with respect to Lebesgue measure, which has many nice properties.

The restriction of the Lebesgue measure to any convex set is also log-concave.

Gaussian measure and Lebesgue measure on the real line are equivalent to one another.

Here is an example of a set of second category in R with Lebesgue measure 0.

The Lebesgue measure defined on all of R has infinite curvature.

The real line carries a canonical measure, namely the Lebesgue measure.

In the continuous univariate case above, the reference measure is the Lebesgue measure.

Lambda denotes the Lebesgue measure in mathematical set theory.

In technical terms, the set of nonregular economies is of Lebesgue measure zero.