EQUIVALENCE OF LEBESGUE-STIELTJES MEASURES GENERATED WITH DISTRIBUTION FUNCTION - Pažanin, Ratomir (Matematichki Bilten, 1996)
This paper discusses the equivalence of Lebesgue – Stieltjes measures \(μ_F\) and \(μ_H\), generated with probability distribution functions \(F\) and \(H=F°φ\) of random variables \(X\) and \(Y=φ^-1° X\). It is proved that measures \(μ_F\) and \(μ_H\) are equivalent in the following sense: \[ μ_H(B)=0 μ_F(B)=0 \] for every \(μ_H\) (i.e. \(μ_F\)) negligible set \(B\) from \(σ\)-algebra \(B\) on \(R\), if every strictly increasing and continuous function \(φ:R→ R\) satisfies condition \[ μ_F(B)=0 μ_F(φ(B))=μ_F(φ^-1(B))=0 \] for every \(μ_F\) negligible set \(BB\). It is shown that conditions which the function \(φ:R→ R\) must satisfy for equivalence of measures \(μ_F\) and \(μ_H\), \(H=F°φ\) are much simplier if the distribution function \(F\), which generates \(L-S\) measure \(μ_F\), is only absolutely continuous, or singular, or discrete. Finally, a singular probability distribution is constructed and a function \(φ(x) e x\) for which the singular measures \(μ_F\) and \(μ_H\), \(H=F°φ\), are equivalent.