Two indistinguishable laws leave error on an unseen atom

Let $\mathcal{X}=\{a,b\}$ with the discrete sigma-algebra and feature marginal $P_X(a)=P_X(b)=1\mathrm{/}2$, and observe only the labeled pair $(a,0)$. Under $P_0$, the labels at $a,b$ are $0,0$; under $P_1$, they are $0,1$. The common learner input consists of this labeled pair and the full marginal $P_X$. Let $\mathcal{H}=\{g:\mathcal{X}\to \{0,1\}\}$, and let $Q:\mathcal{H}\to [0,1]$ satisfy ${\sum }_{g\in \mathcal{H}}Q(g)=1$, selected from that common input, so it is the same under both laws. For $g\in \mathcal{H}$, define $R_{P_0}(g)=\frac{1}{2}\mathbf{1}\{g(a)\mathrm{\ne }0\}+\frac{1}{2}\mathbf{1}\{g(b)\mathrm{\ne }0\},R_{P_1}(g)=\frac{1}{2}\mathbf{1}\{g(a)\mathrm{\ne }0\}+\frac{1}{2}\mathbf{1}\{g(b)\mathrm{\ne }1\}\mathrm{.}$ The expected population risks satisfy ${\max }_{j\in \{0,1\}}{\sum }_{g\in \mathcal{H}}Q(g)R_{P_j}(g)\ge 1\mathrm{/}4$.

Each panel gives feature, label, and mass. The input at $a$ is observed in both panels; the feature masses agree, while only the unobserved label at $b$ changes.

diagram failed to render:
\begin{tikzpicture}
\draw (0,0) rectangle (3.5,1.8);
\node at (1.75,1.45) {$P_0$};
\node at (0.9,0.9) {$a:0$};
\node at (2.6,0.9) {$b:0$};
\node at (0.9,0.35) {$1/2$};
\node at (2.6,0.35) {$1/2$};
\draw (4,0) rectangle (7.5,1.8);
\node at (5.75,1.45) {$P_1$};
\node at (4.9,0.9) {$a:0$};
\node at (6.6,0.9) {$b:1$};
\node at (4.9,0.35) {$1/2$};
\node at (6.6,0.35) {$1/2$};
\end{tikzpicture}