# One point on $\phi$-irreducibility

Let $P(x,A)$ be a stochastic kernel on a measurable space $(E,\mathcal E)$ and $G = \sum\limits_0^\infty P^n$ be its potential kernel. A $\sigma$-finite measure $\phi$ is called the irreducibility measure if $\phi(A)>0$ implies $PG(x,A)>0$ for all $x\in E$.

In Numellin's book it is written that if $\phi$ is an irreducibility measure then so is $\phi P$. I've tried to prove this fact but didn't manage to do it, so I hope you can help me with an idea.

Just to clarify things: kernel $P$ means that $P(x,\cdot)$ is a probability measure on $(E,\mathcal E)$ for all $x$ and $P(\cdot, A)$ is a measurable function for all $E$. The product of kernels, say $P$ and $G$ is $$PG(x,A) = \int\limits_E G(y,A)P(x,\mathrm dy)$$ and $\phi P(A) = \int\limits_E P(y,A)\phi(\mathrm dy)$.

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Well, $\phi P(A) > 0$ implies $\exists B\subset E: \phi(B)>0$ and $P(y,A)>0\ \forall y\in B$. This implies by definition of $\phi$ that $PG(x,B) > 0$ for all $x\in E$, whence $P^2G(x,A) = \int P(y,A) PG(x,dy) > 0$ for all $x\in E$. But $PG(x,A) \ge P^2G(x,A)$.
Thank you very much, Pascal. Correct me if I wrong: we can take $B = \{y:P(y,A) > 0\}$ and so $$P^2 G(x,A) = \int\limits_E P(y,A)PG(x,dy) = \int\limits_B P(y,A)PG(x,dy) >0$$ since $PG(x,B)>0$ for all $x\in E$ from the fact that $\phi(B)>0$. If I may ask - how did you get the intuition about this proof? I feel not very confident in dealing with closed sets and irreducibility so far, so in some problems for me it's hard to get the idea. –  Ilya Apr 27 '12 at 12:57
@Ilya, you are welcome. Yes, you can define $B$ like this. For the intuition: $\phi$ "gives" the set of all states that can be reached from any point. Now if from these states we take one step further, i.e. if we map $\phi$ to $\phi P$, then a fortiriori, we can reach these states from any point. –  Pascal Maillard Apr 28 '12 at 6:58
@Pascal: I've just accepted it - just wanted first to verify the answer by myself to be sure that I've understood in completely. I wonder if you can advise any references with exercises on the topic of $\phi$-irreducibility. Or maybe even references on the application of this theory to the analysis of the concrete models. –  Ilya Apr 28 '12 at 18:48