I encountered here the following definition of recurrence for continuous-state-space Markov chains:
For every measurable set $B$ with $\phi(B)>0$, $$\mathbf P(X_1,X_2,\ldots\in B\text{ i.o.}|X_0) > 0$$ for all $X_0$, and $$\mathbf P(X_1,X_2,\ldots\in B\text{ i.o.}|X_0) = 1$$ for $\phi$ almost-every $X_0$.
Apparently, $\phi$ is a measure, and $X_0,X_1,\ldots$ are random variables. What does "$\phi$ almost-every $X_0$" mean?