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Let $X$ be a topological space with a Borel $\sigma$-algebra $\mathcal B(X)$. There is a family of probability measure on $X$, which is denoted as $P:X\times \mathcal B(X)\to[0,1]$.

I would like to consider a family with a following property:

For any non-empty open set $B$ there exists $x\in X$ such that $P(x,B)>0$.

Roughly speaking, the support of family of measures $\{P(x,\cdot)\}_{x\in X}$ is the whole space $X$. I wonder if it can be said formally. It is not irreducibility of a Markov chain, since the chain can admit this property being reducible and may not admit this property being irreducible.

The support of measure is defined as $ \operatorname{supp}\mu = \{x\in X:\mu(U(x))>0\forall \text{ open neighborhood }U(x) \}. $

I guess that the support of family of measures should be defined like this $ \operatorname{supp}P:=\overline{\bigcup\limits_{x\in X}\operatorname{supp}P(x,\cdot)} $ but first, it's not clear if $\operatorname{supp}P=X$ is equivalent to my condition, second, I had no chance to find this definition of $\operatorname{supp}P$ in the literature and finally this definition seems a bit weird cause there is an infinite union of closed sets.

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    @GeorgeLowther: ok, thank you for the clarification2011-10-08

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