Nonmeasurable images

TL;DR

This paper investigates the nonmeasurability of images of subsets in Polish spaces under certain mappings, addressing a specific question about projections of subsets of the unit disc and their Lebesgue measurability.

Contribution

It demonstrates the relative consistency of nonmeasurable projections for certain sets under multiple continuous functions, extending known results about Bernstein sets.

Findings

Existence of sets with continuum many measurable and non-measurable projections.

Relative consistency of nonmeasurability for certain sets with fewer than continuum many functions.

Extension of Bernstein set properties to nonmeasurable images under continuous mappings.

Abstract

In this article we will investigate nonmeasurability with respect to some $\sigma$-ideals in Polish space $X,$ of images of subsets of $X$ by selected mappings defined on the space $X$. Among of them we answer the following question: "It is true that there exists a subset of the unit disc in the real plane such that the continuum many projections onto lines are Lebesgue measurable and continuum many projections are not?". It is known that there exists continuous function $f:[0,1]\to [0,1]$ such that for every Bernstein set $B\subseteq [0,1]$ we have $f[B]=[0,1].$ We show relative consistency with $ZFC$ of fact that the above result is not true for some $\cn$ or $\cm$-completely nonmeasurable sets, even if we take less than $\c$ many continuous functions.