An uncountable supremum of measurable random variables need not be measurable. Since \(\ell^\infty(\mathcal F)\) is usually nonseparable, even the empirical process viewed as a function-valued map may fail to be Borel measurable. Outer probability allows rigorous convergence statements before a separability argument resolves this issue.
Outer measure and measurable envelopes
Definition. On a probability space \((\Omega,\mathcal A,P)\), the outer probability of an arbitrary set \(A\subseteq\Omega\) is \[ P^\star(A)=\inf\{P(B):A\subseteq B,\ B\in\mathcal A\}. \] For an arbitrary extended-real map \(Z:\Omega\to\overline{\mathbb R}\), its outer expectation is \[ \mathbb E^\star Z =\inf\{\mathbb EY:Y\text{ is measurable and }Y\ge Z\}, \] whenever the positive and negative parts are not both infinite.
Definition (minimal measurable majorant). A measurable \(Z^\star\ge Z\) is a measurable majorant. Under a complete probability space, one can choose \(Z^\star\) so that every measurable majorant \(Y\ge Z\) satisfies \(Y\ge Z^\star\) almost surely. Then \[ \mathbb E^\star Z=\mathbb EZ^\star. \]
Similarly, the maximal measurable minorant \(Z_\star\le Z\) gives inner expectation \(\mathbb E_\star Z=\mathbb EZ_\star\). For an arbitrary event \(A\), \[ P_\star(A)=1-P^\star(A^c). \]
Calculus of outer expectation
Basic properties. For arbitrary maps \(X,Y\) and \(a\ge0\): \[ \begin{aligned} X\le Y&\Longrightarrow \mathbb E^\star X\le\mathbb E^\star Y,\\ \mathbb E^\star(aX)&=a\mathbb E^\star X,\\ \mathbb E^\star(X+Y)&\le\mathbb E^\star X+\mathbb E^\star Y. \end{aligned} \] If \(Y\) is integrable and measurable, then \[ \mathbb E^\star(X+Y)=\mathbb E^\star X+\mathbb EY. \]
Proof. Monotonicity and positive homogeneity follow directly by transforming measurable majorants. If \(X^\star,Y^\star\) are minimal majorants, then \(X^\star+Y^\star\) is a measurable majorant of \(X+Y\), giving subadditivity. When \(Y\) is measurable, a majorant \(Z\ge X+Y\) corresponds one-to-one with the majorant \(Z-Y\ge X\); taking infima of expectations yields equality.
Outer Markov inequality. If \(Z\ge0\), then for every \(t>0\), \[ P^\star(Z>t)\le\frac{\mathbb E^\star Z}{t}. \]
Proof. For every measurable majorant \(Y\ge Z\), \[ \{Z>t\}\subseteq\{Y>t\}, \] so \(P^\star(Z>t)\le P(Y>t)\le\mathbb EY/t\). Take the infimum over \(Y\).
Outer convergence in probability
Definition. For maps \(Z_n,Z\) into a metric space \((S,d)\), \[ Z_n\to_{P^\star} Z \] means \[ P^\star\{d(Z_n,Z)>\varepsilon\}\to0 \quad\text{for every }\varepsilon>0. \] Outer \(o_p(1)\) and \(O_p(1)\) use this probability and outer tightness.
Outer Slutsky principle. If \(X_n=o_{P^\star}(1)\) and \(Y_n=O_{P^\star}(1)\), then \[ X_nY_n=o_{P^\star}(1). \]
Proof. Given \(\varepsilon,\eta>0\), choose \(M\) so that \[ \limsup_nP^\star(|Y_n|>M)<\eta. \] Then \[ P^\star(|X_nY_n|>\varepsilon) \le P^\star(|Y_n|>M) +P^\star(|X_n|>\varepsilon/M). \] Take the limsup and then \(\eta\downarrow0\).
Why suprema can be nonmeasurable
For each \(f\in\mathcal F\), \(\mathbb G_nf\) is measurable. But \[ \|\mathbb G_n\|_{\mathcal F} =\sup_{f\in\mathcal F}|\mathbb G_nf| \] is an uncountable supremum; sigma-fields are closed only under countable unions, so measurability does not follow formally. Likewise, the map \[ \omega\mapsto\{\mathbb G_nf(\omega):f\in\mathcal F\} \] may fail to be Borel measurable into nonseparable \(\ell^\infty(\mathcal F)\).
Pointwise measurable classes
Definition. A class \(\mathcal F\) is pointwise measurable if a countable subclass \(\mathcal F_0\subseteq\mathcal F\) exists such that, for every \(f\in\mathcal F\), there are \(f_m\in\mathcal F_0\) with \[ f_m(x)\to f(x)\qquad\text{for every }x\in\mathcal X. \]
Measurable supremum reduction. If \(\mathcal F\) is pointwise measurable and \(Pf_m\to Pf\) whenever \(f_m\to f\) through the approximating sequence, then \[ \sup_{f\in\mathcal F}|\mathbb G_nf| =\sup_{f\in\mathcal F_0}|\mathbb G_nf|, \] and the supremum is measurable.
Proof. For every sample, \[ P_nf_m\to P_nf \] because the empirical average has only finitely many evaluation points. By assumption \(Pf_m\to Pf\), hence \(\mathbb G_nf_m\to\mathbb G_nf\). Every coordinate in the large class is therefore a limit of coordinates in the countable class, so the two suprema agree. A countable supremum of measurable variables is measurable.
Example. The threshold class \[ \{1_{(-\infty,t]}:t\in\mathbb R\} \] is pointwise measurable after using rational thresholds together with a consistent one-sided convention. Monotonicity of distribution functions then identifies the full supremum with a countable one.
Asymptotic measurability
Definition. A sequence of maps \(Z_n\) into a metric space \(S\) is asymptotically measurable if \[ \mathbb E^\star h(Z_n)-\mathbb E_\star h(Z_n)\to0 \] for every bounded continuous \(h:S\to\mathbb R\).
This condition says that any nonmeasurability becomes asymptotically invisible to bounded continuous tests. Weak convergence in the Hoffmann-Jorgensen sense combines outer expectations with asymptotic measurability. In applications, pointwise measurability, image admissibility, or a separable version of the process usually supplies the condition; otherwise outer notation must remain in theorem statements.