Full Lebesgue measure(Dynamical systems)

Question

I am reading a paper and there is a theorem which says:

The dynamical system $(D,g)$ is called ergodic on $K\subset D$ if for any saturated subset $A\subset D$, its intersection with $K$ is of either zero or full(in $K$) Lebesgue measure.

What does a set of full Lebesgue measure mean?

Thanks a lot

Answer 1

It's mean that or $\mu(A)=0$ or $\mu(A)=\mu(D)=1$, where $\mu$ it's the Lebesgue measure on D and the result is one of the condition for ergodicity of a system. The others are:

1) Equality of time-average and mean value $$\bar{f}(x)=\langle f\rangle_\mu$$

2) Equality of frequency of visit and measure $$\nu_D:=\chi_D(x)=\mu(D)$$

3) Metric indecomposability (your problem) $$\Phi^t(B)=B\quad\forall t\Rightarrow \mu(B)=0\quad or\quad \mu(B)=1$$

4) Absence of first integrals $$f(\Phi^t(x))=f(x)\Rightarrow f(x)=const.$$

Where in 1,2 and 4 the results holds $\mu-$almost everywhere.

It can be show that $1\Rightarrow2\Rightarrow3\Rightarrow1$ and $3\Leftrightarrow4$.

Answer 2

Definition: A set has full measure if and only if its complement has measure zero.

A set has full measure in $K$ if its complement in $K$ (that is, the intersection of its complement with $K$) has measure zero.