Bounded quadrant modification and generator norms

Under [A1], [F8] makes $(\mathbb{R},\mathcal{L},\lambda )$ a complete measure space with the usual interval lengths. The coordinate pairing $⟨z,w⟩={\sum }_j{z}_j\overline{w_j}$ is linear in its first variable and conjugate symmetric by finite sum algebra, and its diagonal is ${\sum }_j\mathrm{\mid }{z}_j{\mathrm{\mid }}^2$, which is nonnegative and vanishes exactly at zero. Its induced length is the displayed Euclidean norm by [F11]. The spaces $V$ and $E$ have their ordered matrix-unit bases. For a matrix unit $e_{ij}$, $\mathrm{\parallel }{e}_{ij}{\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}}=1$ and $\mathrm{\mid }{X}_{ij}\mathrm{\mid }\le \mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}}$, so $\mathrm{\parallel }T(X){\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}}\le {\sum }_{ij}\mathrm{\mid }{X}_{ij}\mathrm{\mid }\mathrm{\parallel }T(e_{ij}){\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}}\le ({\sum }_{ij}\mathrm{\parallel }T(e_{ij}){\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}})\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathrm{s}\mathrm{p}}$. Matrices themselves are bounded on Euclidean space by their finite entry sums. Thus the norms in the Statement are finite operator norms, and [F2] and [F3] make $E$ a finite-dimensional Banach space. Its coordinates $T\mapsto (T(e_{ij}){)}_{kl}$ are continuous, since their absolute values are at most $\mathrm{\parallel }T\mathrm{\parallel }$. For a continuous real coordinate functional $h$ on $E$, the family of Borel target sets whose inverse images under $h$ are norm-Borel is a sigma-algebra containing every open set; hence $h$ is Borel measurable by [F12]. The inverse-image identity for a composition therefore makes every real and imaginary coordinate of $\mathcal{G}$ product-measurable. The coordinate bounds just proved and the finite-basis estimate $\mathrm{\parallel }{\sum }_a{c}_a{H}_a\mathrm{\parallel }\le {\sum }_a\mathrm{\mid }{c}_a\mathrm{\mid }\mathrm{\parallel }{H}_a\mathrm{\parallel }$ from [F2] identify the norm topology of $E$ with its finite real coordinate topology. Every norm-open set is therefore a union of the countably many rational coordinate boxes it contains. Its inverse image under a map with measurable coordinates is a countable union of finite intersections of coordinate preimages. Thus every such map into $E$ is norm-Borel measurable. The swap $S(s,u)=(u,s)$ is measurable because the preimage of a measurable rectangle is the rectangle with its two factors reversed, and these rectangles generate the product sigma-algebra in [F12]. Thus $\mathcal{G}\circ S$ also has measurable coordinates. The finite coordinate sums, differences, products and positive and negative parts used here and below are measurable by [F15]. Apply the section theorem in [F6] to the real and imaginary coordinates of the product-measurable kernel to obtain measurability of each section coordinate at every parameter. For any map into $E$ with measurable coordinates, round every real and imaginary coordinate to a grid of mesh $1\mathrm{/}n$ on the coordinate box $[-n,n]$, and set the approximant to zero outside that box. These are measurable finite-valued simple functions; at each point the coordinates eventually lie in the box and converge, hence converge in norm by the finite basis bound $\mathrm{\parallel }{\sum }_a{c}_a{H}_a\mathrm{\parallel }\le {\sum }_a\mathrm{\mid }{c}_a\mathrm{\mid }\mathrm{\parallel }{H}_a\mathrm{\parallel }$. This proves strong measurability of $\mathcal{G}$, its transposed kernel and all their fixed-time sections, without a separate measurability theorem for Banach-valued maps. Coordinate integration agrees with Bochner integration: for simple functions it is the same finite sum, and for an integrable-simple approximation each coordinate's integral error is bounded by the integral of the norm error.

Besides translation from [F7], reflection preserves Lebesgue measure on the line. Indeed, reflect a half-open interval covering to intervals of the same lengths, and enlarge the reflected intervals to half-open intervals with total extra length less than any prescribed $\epsilon >0$ (use extra lengths at most $\epsilon 2^{-j-1}$ in an enumeration of all component intervals). The elementary-cover definition then gives ${\lambda }^{\ast }(-A)\le {\lambda }^{\ast }(A)+\epsilon$ for every set with finite outer measure; reflecting twice gives equality, also covering the infinite case. Substituting $-C$ for the test set $C$ in the Caratheodory criterion proves that reflection preserves measurable sets and their measure. Therefore translations and reflection preserve integrals of nonnegative measurable functions: their inverse images preserve indicator integrals, then finite nonnegative simple integrals, and finally the supremum defining the nonnegative integral. Consequently, for every $t$, ${\int }_{\mathbb{R}}q(t-u)du={\int }_{\mathbb{R}}q(s-t)ds={\int }_{\mathbb{R}}q=\mathrm{\Gamma }\mathrm{/}2$. The sections in $B(t),L(t)$ are strongly measurable by step 1.1, so [F4] gives their Bochner integrability, and [F5] gives both asserted marginal bounds.

For any nonnegative Lebesgue-measurable $f$ and fixed $s>t$, the one-dimensional substitution $v=s-u$ justified in step 2.1 gives ${\int }_{-\mathrm{\infty }}^{t}f(s-u)du={\int }_{s-t}^{\mathrm{\infty }}f(v)dv$. The coordinate pullback $(s,v)\mapsto f(v)$ is product-measurable: the inverse image of a Borel target set $B$ is the measurable rectangle $\mathbb{R}\times f^{-1}(B)$. Lebesgue measurability contains the one-dimensional Borel sigma-algebra: by [F8] it contains elementary intervals and is a sigma-algebra; every open subset of the line is the countable union of the rational half-open intervals contained in it. Consequently [F12] and [F14] give $\mathcal{B}({\mathbb{R}}^2)=\mathcal{B}(\mathbb{R})\otimes \mathcal{B}(\mathbb{R})\subseteq \mathcal{L}\otimes \mathcal{L}$. The region $D_t=\{(s,v):t<s<t+v\}$ is open and hence product-measurable. Its indicator is measurable because its target preimages are among $\mathrm{\varnothing },D_t,D_t^c,{\mathbb{R}}^2$. By [F15], $(s,v)\mapsto {\mathbf{1}}_{D_t}(s,v)f(v)$ is nonnegative and product-measurable, with value zero outside $D_t$ also if $f(v)$ is infinite. The two real-line Lebesgue spaces are sigma-finite, as they are covered by the finite-measure intervals $[-m,m]$. Tonelli therefore yields ${\int }_t^{\mathrm{\infty }}ds{\int }_{-\mathrm{\infty }}^{t}duf(s-u)={\int }_0^{\mathrm{\infty }}dv{\int }_t^{t+v}dsf(v)={\int }_0^{\mathrm{\infty }}vf(v)dv$. This identity permits infinite values and was obtained without presupposing finiteness or a two-dimensional change of variables.

Reflection from step 2.1 and the sum rule in [F15] make $f(v)=q(v)+q(-v)$ measurable. The coordinate differences of $A(s,u)=\mathcal{G}(s,u)-\mathcal{G}(u,s)$ are measurable by step 1.1 and [F15]. The norm is continuous, since $\mathrm{\mid }\mathrm{\parallel }T\mathrm{\parallel }-\mathrm{\parallel }U\mathrm{\parallel }\mathrm{\mid }\le \mathrm{\parallel }T-U\mathrm{\parallel }$ follows from [F2]. Using the same continuous-map inverse-image argument as in step 1.1, its composition with $A$ is product-measurable. Multiplication by the quadrant indicator and the coordinate positive and negative parts preserve measurability by [F15]. Put $A(s,u)=\mathcal{G}(s,u)-\mathcal{G}(u,s)$ and $f(v)=q(v)+q(-v)$. Then $\mathrm{\parallel }A(s,u)\mathrm{\parallel }\le f(s-u)$, and the identity in step 3.1 gives ${\int }_t^{\mathrm{\infty }}ds{\int }_{-\mathrm{\infty }}^{t}du\mathrm{\parallel }A(s,u)\mathrm{\parallel }\le {\int }_0^{\mathrm{\infty }}v(q(v)+q(-v))dv={\int }_{\mathbb{R}}\mathrm{\mid }v\mathrm{\mid }q(v)dv\le \mathrm{\Gamma }\tau$. The scalar norm of $A$ restricted to the quadrant is product-measurable, so Tonelli identifies this finite iterated integral with its product integral. Each fixed-$s$ inner section is integrable, since its norm integral is bounded by ${\int }_{\mathbb{R}}(q(s-u)+q(u-s))du=\mathrm{\Gamma }$. By step 1.1 and [F4], the inner sections and the product kernel are Bochner integrable. Each coordinate has absolute value at most $\mathrm{\parallel }A\mathrm{\parallel }$. Its real and imaginary positive and negative parts are nonnegative product-measurable; Tonelli makes their section integrals measurable. All are finite for every $s$ by the section bound, so their differences give measurable coordinates of the actual inner-integral map at every $s$. Scalar Fubini applies to these integrable coordinates and identifies their product and iterated integrals. Step 1.1 gives its strong measurability; [F5] bounds its norm by the inner norm integral, and the finite estimate above and [F4] give its outer Bochner integrability. Applying [F5] once more gives $\mathrm{\parallel }M(t)\mathrm{\parallel }\le \mathrm{\Gamma }\tau$. Coordinate Fubini also identifies this iterated integral with the quadrant product integral. Together with step 2.1, this proves all the absolute-convergence and norm claims for every $t$.