Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$).
If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in B}\bigcap_{t\in\mathbb{R}}F_{-h(t)}(B[F_{t}(x),\delta])$$ where $B=\{h:\mathbb{R}\rightarrow \mathbb{R}$ continuous with $h(0)=0\}$ i.e. $z\in \Lambda(x,\delta)$ sss $\exists \ h:\mathbb{R}\rightarrow \mathbb{R}$ continuous with $h(0)=0$ and $d(F_{t}(x),F_{h(t)}(z))\leq \delta$ for all $t\in \mathbb{R}$.
Let $M(X)$ the set of measures of probability borel in $X$ (with topology weak is compact metric space), we define
$E=\{\mu\in M(X): \ \exists \delta>0 \ \mbox{such that }\ \mu(\Lambda(x,\delta))=0$ for all $x\in X$ }
I would like some example of flow $F$ for which the set $ E $ is not a set of Baire in $M(X)$.
I appreciate any suggestions.