The following excerpt is taken from Shen and Wasserman (2001).
I have difficulty understanding some terminologies. On line 4,
[...] each $P_\eta$ is a probability on $(\mathscr Y,\mathscr B)$ with density $p(y\mid\eta)$ with respect to a common, dominating, $\sigma$-finite measure $\lambda$.
In particular, what do the authors mean by saying a probability (distribution) is with respect to a $\sigma$-finite measure?
Suppose $\mu$ is a measure on a $\sigma$-field, and $f$ is a $\mu$-measurable function, and $f\ge 0$ everywhere (or almost everywhere, if you like). For every $\mu$-measurable set $A$, let $$ \nu(A) = \int_A f\,d\mu. $$ Then $f$ is the density of the measure $\nu$ with respect to the measure $\mu$. It's also called the Radon–Nikodym derivative, and one writes $f=\dfrac{d\nu}{d\mu}$.
For example, if $\mu$ is Lebesgue measure on the line $\mathbb R$, then the measure that $\mu$ assigns to every interval is its length. If $\nu$ is the standard normal distribution on the line, then density of $\nu$ with respect to $\mu$ is $$ \frac{d\nu}{d\mu}(x) = \frac 1 {\sqrt{2\pi}} e^{-x^2/2}. $$
As another example, suppose $\mu$ is "counting measure" on the set $\{0,1,2,3,4\}$, so that, for example, $\mu(\{1,2,4,6\})=4$. Then the density with respect to $\mu$ of the binomial distribution with parameters $4$ and $1/2$ is $f(x) = \dbinom 4 x (1/2)^4$.