Extension of Likelihood-density

Question

Given an observation scheme, say $(X_1,X_2,\ldots,X_n)$ where $X_i$ evolves to one measure, one can define the likelihood function, a parameter dependend density of the form $$ L_n:=L(X_1,\ldots,X_n\mid\theta) $$ This is a density wrt to a measure say $P^n$. Thus there exists a measure e $Q^n$, such that $L_n$ is the radon-nikodym-derivative. However, for assumptions about consistency of an estimator one lets $n\rightarrow \infty$ can we expect a measure $Q^{\infty}?$

Answer 1

Edit: I misinterpreted "consistency of an estimator" to be referring to Kolmogorov's extension theorem, as opposed to statistical consistency of an estimator, so what follows below is not right.

Usually the likelihood function factors into $L_n:=L(X_1\mid\theta)\cdots L(X_n\mid\theta)$.

In this case the associated measure of likelihood function on $Q^\infty$ will exist by virtue of the Kolmogorov extension theorem, so long as your finite measure satisfies consistency conditions listed in the link.