PUSHFORWARD MEASURE

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

Contents

Definition

Examples and applications

Definition

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
:$(f\_\{$
★ } (mu)) (B) = mu left( f^{-1} (B)
ight) mbox{ for } B in Sigma_{2}.
This definition applies ''mutatis mutandis'' for a signed or complex measure.

Examples and applications

★ A natural "Lebesgue measure" on the unit circle 'S'¹ (here thought of as a subset of the complex plane 'C') may be defined using a push-forward construction and Lebesgue measure ''λ'' on the real line 'R'. Let ''λ'' also denote the restriction of Lebesgue measure to the interval [0, 2''π'') and let ''f'' : [0, 2''π'') → 'S'¹ be the natural bijection defined by ''f''(''t'') = exp(''i'' ''t''). The natural "Lebesgue measure" on 'S'¹ is then the push-forward measure ''f''_∗(''λ''). The measure ''f''_∗(''λ'') might also be called "arc length measure" or "angle measure", since the ''f''_∗(''λ'')-measure of an arc in 'S'¹ 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 torus 'T'^''n''. The previous example is a special case, since 'S'¹ = 'T'¹. This Lebesgue measure on 'T'^''n'' is, up to normalization, the Haar measure for the compact, connected Lie group 'T'^''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 functional in the continuous dual space to ''X'' is a Gaussian measure on 'R'.

★ Consider a measurable function ''f'' : ''X'' → ''X'' and the composition of ''f'' with itself ''n'' times:
::$f^\{(n)\}\; =\; underbrace\{f\; circ\; f\; circ\; dots\; circ\; f\}\_\{n\; mathrm\{,\; times\}\}\; :\; X\; o\; X.$
: 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.