Partially one-homogeneous signed measure projection

The coordinate $w$ is measurable, so $w_{+}$ and $w_{-}$ are nonnegative measurable functions. By [F1], ${\alpha }_{\mu }$ and ${\beta }_{\mu }$ are measures, and [F2] gives ${\alpha }_{\mu }(\mathrm{\Theta }),{\beta }_{\mu }(\mathrm{\Theta })\le \int \mathrm{\mid }w\mathrm{\mid }d\mu <\mathrm{\infty }$.

For each measurable $B$, $w=w_{+}-w_{-}$ and both integrals are finite, so ${\int }_{\mathbb{R}\times B}wd\mu ={\alpha }_{\mu }(B)-{\beta }_{\mu }(B)$. Countable additivity of both finite measures makes their difference countably additive with finite real values, hence a finite signed measure by [F3]. For every $B\in \mathcal{T}$ and every countable measurable partition $(B_j)$ of $B$, ${\sum }_j\mathrm{\mid }{h}^1(\mu )(B_j)\mathrm{\mid }\le {\sum }_j({\alpha }_{\mu }(B_j)+{\beta }_{\mu }(B_j))={\alpha }_{\mu }(B)+{\beta }_{\mu }(B)$; taking the supremum in [F4] proves the stated pointwise total-variation bound. The weighted penalty inequality follows by monotonicity of the nonnegative integral, first for simple nonnegative weights and then by increasing simple approximation. For a finite-valued simple $\phi$, linearity [F5] and the definitions of ${\alpha }_{\mu },{\beta }_{\mu }$ give $\int \phi d{\alpha }_{\mu }-\int \phi d{\beta }_{\mu }=\int w\phi d\mu$. For general bounded measurable $\phi$, choose $M$ with $\mathrm{\mid }\phi \mathrm{\mid }\le M$ and partition $[-M,M]$ into finitely many intervals of length at most $1\mathrm{/}n$; assigning one endpoint on each measurable inverse image gives a finite-valued simple ${\phi }_n$ with $\mathrm{\parallel }{\phi }_n-\phi {\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\mathrm{/}n$. The errors in the two ${\alpha }_{\mu },{\beta }_{\mu }$ integrals are at most $({\alpha }_{\mu }(\mathrm{\Theta })+{\beta }_{\mu }(\mathrm{\Theta }))\mathrm{/}n$, and the error in the $w\phi$ integral is at most $\int \mathrm{\mid }w\mathrm{\mid }d\mu \mathrm{/}n$. Letting $n\to \mathrm{\infty }$ proves the identity.

Let $\phi :\mathrm{\Theta }\to \mathcal{H}$ have the bounded everywhere simple approximation specified in the Definition. Its approximants pulled back to the product, multiplied by scalar simple functions converging everywhere to $w$, show that $w\phi (\theta )$ is strongly measurable; moreover $\mathrm{\parallel }w\phi (\theta )\mathrm{\parallel }\le \mathrm{\mid }w\mathrm{\mid }\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$, so [F6] makes it Bochner integrable. For each $z\in \mathcal{H}$, [F7] moves the bounded functional $x\mapsto ⟨z,x⟩$ inside that Bochner integral, and step 2.1 identifies the resulting scalar integral with the pairing in the Definition. Equality of all inner products proves the displayed feature-moment factorization.