Lower bounds for average sample size in the tests of invariability

This article is a continuation of a lower bounds research for average sample size in specific procedures of statistical inference (see [1] and [2]). Let $\mathscr P$ be a family of absolutely continuous distributions $P$ on a measurable space $\langle\mathscr X,\mathscr A\rangle$ and $\mathfrak S$ is a group of transformations of $\mathscr X$. The problem of testing hypothesis on an invariance (with respect to $\mathfrak S$) of distribution $P$ of the observed random variable is considered. The hypothesis is a statement
$$ P\in\{P:P(A)=P(sA)\ \forall\,A\in\mathscr A,\ \forall\,s\in\mathfrak\sigma\}, $$
the alternative is $P\in\{P:\exists\,t\in\mathfrak S,\ \sup_A|P(A)-P(tA)|\ge\Delta\}$. The problem of verifying the homogeneity of distributions $P_1$ and $P_2$ of two random variables is also considered. In this case the hypothesis is a statement
$$ (P_1,P_2)\in\{(P_1,P_2):\exists\,t\in\mathfrak S,\ P_1(A)=P_2(tA)\ \forall\,A\in\mathscr A\}, $$
the alternative is $(P_1,P_2)\in\{(P_1,P_2):\sup_A|P_1(A)-P_2(sA)|\ge\Delta\ \forall\,s\in\mathfrak S$. Lower bounds for average sample size which is sufficient for the acceptance of decition on the propound hypothesis trustiness with guaranteed restrictions $(\alpha,\beta)$ on the probabilities of errors is established.