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# Proof of Karlin-Rubin Theorem

Regarding the proof of Casella and Berger Theorem 8.3.17 (Karlin-Rubin), I have two questions.

To use Corollary 8.3.13 (Neyman Pearson with sufficient statistic), our test in the theorem (rejecting $H_0$ if $T > t_0$, denoted by power function $\beta$) has to be equivalent to

\begin{equation}

\begin{aligned}

\text{reject $H_0$ if } \frac{g(T|\theta')}{g(T|\theta_0)} > k'

\end{aligned}

\end{equation}

For the justification, the book has a line $T > t_0 \iff \frac{g(t|\theta')}{g(t|\theta_0)} > k'$. But this is an error; the errata of the book says this line has to be replaced with

\begin{equation}

\begin{aligned}

\left\{ t: \frac{g(t|\theta')}{g(t|\theta_0)} > k' \right\} \subset \{ t: t>t_0\} \subset \left\{ t: \frac{g(t|\theta')}{g(t|\theta_0)} \ge k' \right\}.

\end{aligned}

\end{equation}

My first question is: how is this subset relationship used to establish that the test $\beta$ is equivalent to rejecting $H_0$ if $\frac{g(T|\theta')}{g(T|\theta_0