  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.
