- Radonifying function
In
measure theory , a radonifying function (ultimately named afterJohann Radon ) betweenmeasurable space s is one that takes acylinder set measure (CSM) on the first space to a true measure on the second space. It acquired its name because thepushforward measure on the second space was historically thought of as aRadon measure .Definition
Given two
separable Banach space s E and G, a CSM mu_{T} | T in mathcal{A} (E) } on E and acontinuous linear map heta in mathrm{Lin} (E; G), we say that heta is "radonifying" if the push forward CSM (see below) left{ left. left( heta_{*} (mu_{cdot}) ight)_{S} ight| S in mathcal{A} (G) ight} on G "is" a measure, i.e. there is a measure u on G such that::left( heta_{*} (mu_{cdot}) ight)_{S} = S_{*} ( u)for each S in mathcal{A} (G), where S_{*} ( u) is the usual push forward of the measure u by the linear map S : G o F_{S}.Push forward of a CSM
Because the definition of a CSM on G requires that the maps in mathcal{A} (G) be
surjective , the definition of the push forward for a CSM requires careful attention. The CSM::left{ left. left( heta_{*} (mu_{cdot}) ight)_{S} ight| S in mathcal{A} (G) ight}is defined by::left( heta_{*} (mu_{cdot}) ight)_{S} = mu_{S circ heta}if the composition S circ heta : E o F_{S} is surjective. If S circ heta is not surjective, let ilde{F} be the image of S circ heta, let i : ilde{F} o F_{S} be theinclusion map , and define::left( heta_{*} (mu_{cdot}) ight)_{S} = i_{*} left( mu_{Sigma} ight),where Sigma : E o ilde{F} (so Sigma in mathcal{A} (E)) is such that i circ Sigma = S circ heta.ee also
*
Abstract Wiener space
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