Pushforward measure

In measure theory, a discipline within mathematics, 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. In measure theory, a discipline within mathematics, 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. Given measurable spaces ( X 1 , Σ 1 ) {displaystyle (X_{1},Sigma _{1})} and ( X 2 , Σ 2 ) {displaystyle (X_{2},Sigma _{2})} , a measurable mapping f : X 1 → X 2 {displaystyle fcolon X_{1} o X_{2}} and a measure μ : Σ 1 → [ 0 , + ∞ ] {displaystyle mu colon Sigma _{1} o } , the pushforward of μ {displaystyle mu } is defined to be the measure f ∗ ( μ ) : Σ 2 → [ 0 , + ∞ ] {displaystyle f_{*}(mu )colon Sigma _{2} o } given by This definition applies mutatis mutandis for a signed or complex measure.The pushforward measure is also denoted as μ ∘ f − 1 {displaystyle mu circ f^{-1}} , f ♯ μ {displaystyle f_{sharp }mu } , f ♯ μ {displaystyle fsharp mu } , or f # μ {displaystyle f#mu } . Theorem: A measurable function g on X2 is integrable with respect to the pushforward measure f∗(μ) if and only if the composition g ∘ f {displaystyle gcirc f} is integrable with respect to the measure μ. In that case, the integrals coincide, i.e., In general, any measurable function can be pushed forward, the push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator. In finite-dimensional spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure. The adjoint to the push-forward is the pullback; as an operator on spaces of functions on measurable spaces, it is the composition operator or Koopman operator.