arXiv:2601.09564v2 Announce Type: replace Abstract: The minimum Type II error probability (or volume) of simple hypothesis tests with randomized detectors is expressed as a Riemann integral of the (cumulative) distribution function of the likelihood ratio for all non-negative Type I error probability (or volume) values: $\beta(\varepsilon)=\int_{0}^{\infty}|F(1/\tau)-\varepsilon|^{+}d\tau$ for all $\varepsilon\geq0$. The derivation relies on convex conjugation (the Legendre transform) and level set integration, and is sufficiently general to extend to tests between $\sigma$-finite measures. Measure change identities relating the $\beta(\cdot)$ functions of different hypothesis testing problems are established. Approximations for Type II and Type I volumes are derived under two different hypotheses concerning the Kolmogorov distance to normality for the log-likelihood ratio distributions. In both the Central Limit Theorem and the large deviations regimes, the resulting non-asymptotic expressions recover and extend state-of-the-art characterizations of optimal performance in the memoryless case, and improve and generalize them in the Markovian case. Finally, The distinction between the implications of the Central Limit Theorem and the Berry--Esseen Theorem for the asymptotic simple hypothesis testing problem is clarified.