pushforward measure

Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function.

It acquired its name because the pushforward measure on the second space was historically thought of as a Radon measure.

Pushforward measure: the measure induced on the target measure space by a measurable function.

The finite-dimensional distributions of are the pushforward measures , where , , is any measurable function.

This is called the pushforward measure The support of a random variable is defined to be the topological support of this measure, i.e.

Intuitively, a perfect measure μ is one for which, if we consider the pushforward measure on the real line R, then every measurable set is "μ-approximately a Borel set".

In measure theory, a pushforward measure (also push forward, push-forward or image measure ) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.

Theorem : A measurable function g on X is integrable with respect to the pushforward measure f(μ) if and only if the composition is integrable with respect to the measure μ.