On Measure Invariance for a 2-valued Transformation

TL;DR

This paper investigates measure invariance for a family of 2-valued transformations on the interval [0,1], establishing a criterion that links parameters of the transformation, measure density, and weight functions.

Contribution

It provides a new criterion for measure invariance in 2-valued transformations involving parameters, measure density, and weight functions.

Findings

Derived a criterion relating parameters for measure invariance

Established conditions for measure invariance with respect to transformation parameters

Linked measure density, transformation parameters, and weight functions

Abstract

We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. We endow S with a set of weight functions $\alpha=\{\alpha_1(x),\alpha_2(x)\}$ and find a criterion of measure invariance under the transformation. This criterion relates the three parameters $a$, $p$, $\alpha$ to each other.