EQUIVALENCE OF LEBESGUE-STIELTJES MEASURES GENERATED WITH DISTRIBUTION FUNCTION

Abstract

This paper discusses the equivalence of Lebesgue – Stieltjes measures \(\mu_F\) and \(\mu_H\), generated with probability distribution functions \(F\) and \(H=F\circ\varphi\) of random variables \(X\) and \(Y=\varphi^{-1}\circ X\). It is proved that measures \(\mu_F\) and \(\mu_H\) are equivalent in the following sense:

\[
\mu_H(B)=0 \Leftrightarrow \mu_F(B)=0
\]

for every \(\mu_H\) (i.e. \(\mu_F\)) negligible set \(B\) from \(\sigma\)-algebra \(\mathcal{B}\) on \(R\), if every strictly increasing and continuous function \(\varphi:R\to R\) satisfies condition

\[
\mu_F(B)=0 \Rightarrow \mu_F(\varphi(B))=\mu_F(\varphi^{-1}(B))=0
\]

for every \(\mu_F\) negligible set \(B\in\mathcal{B}\).

It is shown that conditions which the function \(\varphi:R\to R\) must satisfy for equivalence of measures \(\mu_F\) and \(\mu_H\), \(H=F\circ\varphi\) are much simplier if the distribution function \(F\), which generates \(L-S\) measure \(\mu_F\), is only absolutely continuous, or singular, or discrete.

Finally, a singular probability distribution is constructed and a function \(\varphi(x)\ne x\) for which the singular measures \(\mu_F\) and \(\mu_H\), \(H=F\circ\varphi\), are equivalent.