mathematics, a pushforward measure (also push forward or push-forward) is obtained by transferring ("pushing forward") a measure from one measurable spaceto another using a measurable function.
Given measurable spaces ("X"1, Σ1) and ("X"2, Σ2), a measurable function "f" : "X"1 → "X"2 and a measure "μ" : Σ1 → [0, +∞] , the pushforward of "μ" is defined to be the measure "f"∗("μ") : Σ2 → [0, +∞] given by
This definition applies "
mutatis mutandis" for a signed or complex measure.
Examples and applications
* A natural "
Lebesgue measure" on the unit circleS1 (here thought of as a subset of the complex planeC) may be defined using a push-forward construction and Lebesgue measure "λ" on the real lineR. Let "λ" also denote the restriction of Lebesgue measure to the interval [0, 2"π") and let "f" : [0, 2"π") → S1 be the natural bijection defined by "f"("t") = exp("i" "t"). The natural "Lebesgue measure" on S1 is then the push-forward measure "f"∗("λ"). The measure "f"∗("λ") might also be called " arc lengthmeasure" or "angle measure", since the "f"∗("λ")-measure of an arc in S1 is precisely is its arc length (or, equivalently, the angle that it subtends at the centre of the circle.)
* The previous example extends nicely to give a natural "Lebesgue measure" on the "n"-dimensional
torusT"n". The previous example is a special case, since S1 = T1. This Lebesgue measure on T"n" is, up to normalization, the Haar measurefor the compact, connected Lie groupT"n".
Gaussian measures on infinite-dimensional vector spaces are defined using the push-forward and the standard Gaussian measure on the real line: a Borel measure"γ" on a separable Banach space"X" is called Gaussian if the push-forward of "γ" by any non-zero linear functionalin the continuous dual spaceto "X" is a Gaussian measure on R.
* Consider a measurable function "f" : "X" → "X" and the composition of "f" with itself "n" times:
: This forms a measurable
dynamical system. It is often of interest in the study of such systems to find a measure "μ" on "X" that the map "f" leaves unchanged, a so-called invariant measure, one for which "f"∗("μ") = "μ".
* One can also consider
quasi-invariant measures for such a dynamical system: a measure "μ" on "X" is called quasi-invariant under "f" if the push-forward of "μ" by "f" is merely equivalent to the original measure "μ", not necessarily equal to it.
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