Compare measures of two sets

Question

I am struggling with probably something very trivial. Consider a dynamical system whose evolution equation is given by:

$x^+ = f(x)$,

where $x,x^+\in R^n$ and $f(x)\in R^n$ is piecewise continuous in $x$. If we consider a ball of radius $R$ denoted by $B_R = \{x\in R^n:\|x\|<R\}$ and the evolution set $B_{R^+}=\{x^+\in R^n:x\in B_R\}$, then is the following claim true?

If $\int_{B_R}\|x^+\|dx < \int_{B_R}\|x\|dx$, then Lebesgue measure $\mu(B_{R^+})<\mu(B_{R})$.

Thanks,

Mayank

Answer 1

In general the answer is no.

Take for example $n=1$, $f(x)=x|x|$ and $R=1$. Then $\int_{B_1}|x^+|dx < \int_{B_1}|x|dx$, but $B_{1^+}=B_1$.