In the book "Markov Chains and Stochastic stability" by Meyn and Tweedie the measurable space $(\mathsf X,\mathcal{B}(\mathsf X))$ is said to be $\varphi$-irreducible for a Markov Chain $X$ if there exists a measure $\varphi$ on $\mathcal B(\mathsf X)$ such that $\varphi(A)>0$ implies $L(x;A)>0$ for any $x\in\mathsf X,A\in\mathcal B(\mathsf X)$. Here $L(x;A) = \mathsf P\{\text{there is }k\ge1\text{ such that }X_k\in A|X_0 = x\}.$
I wonder if they mean only the measures $\varphi(\mathsf X)>0$ (i.e. non-trivial). The book is from 1993 and I didn't find there a convention that all measures are assumed to be non-trivial. Moreover, I saw that sometimes they assume non-triviality of some measures explicitly in the text (e.g. the definition of small sets and further in Chapter 5).
My doubts are the following. If $\varphi(\mathsf X) = 0$ then clearly it is an irreducibility measure. Moreover, as in Proposition 4.2.2 it's written that the maximal irreducibility measure $\psi$ is equivalent to \psi'(A) = \int\limits_\mathsf X K_{a_{\frac12}}(y,A)\varphi(dy) for any finite irreducibility measure $\varphi$ for $ K_{a_\frac12}(y,A) = \sum\limits_{n=0}^\infty \frac{1}{2^{n+1}}\mathsf P\{X_n\in A|X_0=y\}. $ Clearly, any trivial measure is finite (unless there is a special convention) and hence the maximal irreducibility measure $\psi$ is always trivial which of course is not true.
I also wonder if there Markov Chains which are not $\psi$-irreducible. The confusion here is that $\psi$ is not a fixed measure, but a maximal irreducible measure which depends on Markov Chain.