In my class on ergodic theory there was a theorem, that all ergodic $F$-invariant measures $\mu$ for a Borel-function $F: X \rightarrow X$ are extreme points of the set $M_F(X) =$ {all $F$-invariant probability-Borel-measures $\mu$ on X}. Vice versa, all $\mu \in ex(M_F(X))$ are ergodic measures. Finally, the extreme points of probability-Borel-measures on X are all Dirac measures $\delta_x, x\in X$. (The extreme points of $M_F(X)$ should only be Dirac-measures as well, right?)
My question is, how can it be that e.g. the Lebesgue-Measure is ergodic for $T_2: [0, 1]\rightarrow [0, 1]$, where $T_2$ is the tent map defined by
\begin{equation}
T_2(x) =
\left\{
\begin{array}{lr}
2x, & \text{if } x \in [0, 1/2]\\
2-2x, & \text{if } x\in [1/2, 1]
\end{array}
\right\},
\end{equation}
since the Lebesgue measure is not a Dirac-Measure, but a $T_2$-invariant probability-Borel-measure on $[0,1]$.
I'm sure I'm missing something obvious but maybe someone can help me point it out.
Best Answer
The extreme points of a subset are not necessarily extreme when considered in the superset.
The extreme points of a unit square are the corners: you can take a triangle as a subset, embedded small enough that it doesn’t touch any sides. The extreme points of this subset are the corners of the triangle, which are no longer extreme when considered in the larger square.