A continuous likelihood-ratio test

Given: The interval data space $([0,1],\mathcal{A})$, where $\mathcal{A}$ is the trace of the Lebesgue sigma-algebra, restricted Lebesgue measure $\mu$, the densities $p_0(x)=1$ and $p_1(x)=2x$, and $\alpha =1\mathrm{/}4$.

[F0] Statistical experiments: an experiment consists of a data measurable space, a parameter set, and one probability law for each parameter.

[A1] The Axiom of Choice: AC is assumed for the general Neyman–Pearson theorem; the example's density representatives $p_0=1$ and $p_1=2x$ are explicit.

[A2] The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): CC is used for the restricted Lebesgue-measure construction and the Riemann-to-Lebesgue equality below.

[F1] Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume, and A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: under CC, restricted Lebesgue measure is a measure; interval mass is length and singleton mass is zero.

[F2] Continuous functions on Euclidean spaces are Borel measurable and A measurable function between measurable spaces: the continuous extension $x\mapsto 2x$ on $\mathbb{R}$ is Borel measurable, so its restriction to $[0,1]$ is measurable for the trace sigma-algebra $\mathcal{A}$.

[F3] The measure with density $f$ relative to $\mu$ and Integrating against a density agrees with integrating the product: $A\mapsto {\int }_{A}2xd\mu$ is a measure and integrates nonnegative $g$ as $\int g(x)2xd\mu (x)$.

[F4] A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: the continuous function $2x$ is Riemann integrable on every compact subinterval.

[F5] Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, For bounded $f$ on $[a,b]$ and a partition $P$: the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$-th subinterval, and the lower and upper Darboux sums $L(f,P)={\sum }_i{m}_i{\mathrm{\Delta }}_i$ and $U(f,P)={\sum }_i{M}_i{\mathrm{\Delta }}_i$, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as ${\sup }_{P}L(f,P)$ and ${\inf }_{P}U(f,P)$, Darboux integrability as their equality, and the notation ${\int }_a^{b}f$, and The Darboux and Riemann definitions agree: a bounded $f$ on $[a,b]$ is Darboux integrable with integral $I$ if and only if for every real $\epsilon >0$ there is a real $\delta >0$ such that $\mathrm{\mid }S(f,P,\xi )-I\mathrm{\mid }<\epsilon$ for every tagged partition of mesh below $\delta$: the bounded Riemann-integrable function has the same Darboux integral, and each lower sum is at most it while each upper sum is at least it.

[F6] Finite sums and finite products, by recursion, Laws of finite sums and finite products, The canonical natural $\iota (n)=n\cdot 1_F$ of a field, and The principle of mathematical induction: finite sums obey their recursion, naturals are interpreted in $\mathbb{R}$, and induction holds.

[F7] For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$: for each $\epsilon >0$ a positive natural $N$ can be chosen with $1\mathrm{/}N<\epsilon$.

[F8] The measure with density $f$ relative to $\mu$, Integrating against a density agrees with integrating the product, and The expectation of an indicator is the probability of the event: density integrals give the corresponding law masses and rejection probabilities.

[F9] Likelihood-ratio tests with boundary randomization, The likelihood-ratio rule is a measurable test of exact size, and Test size and power: the candidate rejects above its likelihood-ratio cutoff and has the displayed null size.

[F10] Neyman–Pearson lemma with singular alternative mass: no test of size at most $1\mathrm{/}4$ has greater power than the candidate.

[F11] Finite, sigma-finite, and semifinite measures: a finite measure is sigma-finite by the one-set cover consisting of the whole space.

[F12] A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral: under CC, a bounded Riemann-integrable function is Lebesgue measurable and its Lebesgue and Riemann integrals agree.

[F13] Probability measures and probability spaces: a measure with total mass $1$ is a probability measure.

[F14] Nonnegative simple measurable functions, The integral of a nonnegative simple function, and The nonnegative integral agrees with the simple integral on simple functions: for every measurable $A$, ${\int }_{A}1d\mu =\mu (A)$.

Proof technique: direct.