Limit of integral over iterated image tending to $0$

Question

Let $f:\mathbb{R}^m\to\mathbb{R}^m$ be a difeomorphism such that $f(B)\subset B$, where $B$ is the unit closed ball and $|\det f'(x) |<1,\,\forall x \in B$. Then, if $g:B\to\mathbb{R}$ is any continuous function, show that: $$\lim_{n\to \infty}\int_{f^n(B)}g(x)dx=0$$ I'm attempting to use the change of variables formula, but I'm having trouble since we dont have that $f$ is $C^1$ and thus we can't guarantee $|\det f'(x) |$ attains maximum, say $\lambda<1$... That is, we only have that: $$|\det (f^{n})'(x) |=|\det f'(f^{n-1}(x)) |\dots|\det f'(x) |<1$$ So that $\int_{f^n(B)}g(x)dx = \int_B g\circ f^n(x)|\det (f^{n})'|dx<\int_B g\circ f^n(x) dx$...Where do I go from here?

Answer 1

I think the proof has to go something like this. If $F\subset \mathbb{R}^m$ let $|F|$ be the Lebesgue measure of $F$. Since $g$ is continuous and $B$ is compact, $$ -\infty < (\min_B g) |f^n (B)| \leq \int_{f^n (B)} g \leq (\max_B g) |f^n (B)| < \infty, $$ so we only need show $\lim |f^n (B)| = 0$.

Let $A_n = f^n (B)$. Then $|A_1| < \infty$ and $A_{n+1} \subset A_n$ for all $n$. It follows $$ \lim |A_n| = \left|\bigcap A_n\right|. $$

Let $E = \bigcap A_n$ and suppose $|E|\neq 0$. It seems to me that $E = f(E)$ and most importantly that $E \subset f(E)$: for if $y\in E$ and we let $x = f^{-1}(y)$ then since $y \in A_n$ for all $n\geq 2$ then $f^{-1}(y) \in A_{n-1}$ for all $n\geq 2$ so $x \in E$. Consequently, $$ |E| \leq |f(E)|, $$ but by assumption $$ |f(E)| = \int_{f(E)} 1 = \int_{E} |\det(Df)| < \int_{E} 1 = |E|, $$ which is a contradiction.

It must be that $|E| = \lim |f^{n} (B)| = 0$ after all, which is what we desired.

Remarks: (1) I learned real analysis from Folland where C^1 is required for the change of variable theorem, however, there is a version that applies here that can be found in Rudin's Real and Complex Analysis. (2) I don't always see 'this set is included in that set' type arguments so clearly, so you should double check these parts! (3) You tag your question multivariable-calculus and Riemann-integration, so probably this is not the approach you are looking for. It is possible it could be re-worked in terms of the multivariable Riemann integral but I will stop short of doing so.