Relaxed objective for a partially one-homogeneous projection

Let $\alpha ,\beta$ be the raw measures in [F1]. For every countable measurable partition $(B_j)$ of a measurable set $B$, ${\sum }_j\mathrm{\mid }\nu (B_j)\mathrm{\mid }={\sum }_j\mathrm{\mid }\alpha (B_j)-\beta (B_j)\mathrm{\mid }\le \alpha (B)+\beta (B)$. Taking the supremum in [F3] gives $\mathrm{\mid }\nu \mathrm{\mid }(B)\le (\alpha +\beta )(B)$. For a nonnegative simple $\tilde{V}$, the definitions of $\alpha ,\beta$ give $\int \tilde{V}d(\alpha +\beta )=\int \mathrm{\mid }w\mathrm{\mid }\tilde{V}(\theta )d\mu$. If $M$ bounds $\tilde{V}$, partition $[0,M]$ into finitely many intervals of length at most $1\mathrm{/}n$ and assign an endpoint on each measurable inverse image. This gives nonnegative simple $V_n$ with $\mathrm{\parallel }{V}_n-\tilde{V}{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}n$. The error on the left is at most $(\alpha (\mathrm{\Theta })+\beta (\mathrm{\Theta }))\mathrm{/}n$, and the error on the right is at most $\int \mathrm{\mid }w\mathrm{\mid }d\mu \mathrm{/}n$; these bounds agree because $\alpha (\mathrm{\Theta })+\beta (\mathrm{\Theta })=\int \mathrm{\mid }w\mathrm{\mid }d\mu$. Taking $n\to \mathrm{\infty }$ proves the identity for $\tilde{V}$. Monotonicity [F4] and the measure domination, with $\mathrm{\mid }\nu \mathrm{\mid }$ a measure by [F7], give $\int \tilde{V}d\mathrm{\mid }\nu \mathrm{\mid }\le \int \tilde{V}d(\alpha +\beta )=\int \mathrm{\mid }w\mathrm{\mid }\tilde{V}(\theta )d\mu$.

The pointwise simple approximation in the Definition makes $\phi$ strongly measurable for the finite measure $\lambda ={\alpha }_{\mu }+{\beta }_{\mu }+{\nu }^{+}+{\nu }^{-}$. Multiplying those approximants by scalar simple functions converging pointwise to $w$ gives a pointwise simple approximation to $w\phi (\theta )$ on the product. Its norm is bounded by $\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }w\mathrm{\mid }$, so both this weighted feature and $\phi$ are Bochner integrable for the relevant measures by [F6]. For every $z\in \mathcal{H}$, the function ${\psi }_z(\theta )=⟨z,\phi (\theta )⟩$ is bounded and measurable. The pairing [F1] and equality $h^1(\mu )=\nu$ give $\int w{\psi }_z(\theta )d\mu =\int {\psi }_{z}d\nu$. By [F2] and [F5], these are respectively $⟨z,\int w\phi d\mu ⟩$ and $⟨z,m_{\nu }⟩$, so equality for every $z$ proves equality of feature moments. Adding the risk term to step 1.1 proves $F(\mu )\ge J(\nu )$. The section maps $\theta \mapsto (1,\theta )$ and $\theta \mapsto (-1,\theta )$ are measurable: preimages of measurable rectangles are either empty or measurable subsets of $\mathrm{\Theta }$. Their preimages preserve disjoint unions, so the displayed formula defines a finite nonnegative measure, of mass ${\nu }^{+}(\mathrm{\Theta })+{\nu }^{-}(\mathrm{\Theta })=\mathrm{\mid }\nu \mathrm{\mid }(\mathrm{\Theta })$. The definition of $h^1$ gives $h^1({\mu }_{\nu })={\nu }^{+}-{\nu }^{-}=\nu$ by [A1]. Direct integration on the two sections gives $\int \mathrm{\mid }w\mathrm{\mid }\tilde{V}d{\mu }_{\nu }=\int \tilde{V}d{\nu }^{+}+\int \tilde{V}d{\nu }^{-}=\int \tilde{V}d\mathrm{\mid }\nu \mathrm{\mid }$ and $\int w\phi d{\mu }_{\nu }=m_{\nu }$. Hence $F({\mu }_{\nu })=J(\nu )$. This proves the infimum identity and its attainment.