Spatial Lq classes satisfy the scalar mixed-norm hypotheses

The displayed functionals are unchanged by almost-everywhere changes of representatives; positivity and homogeneity follow from the corresponding modulus properties. For finite $q$, the triangle inequality follows by applying Minkowski to the two-point counting parameter space; for $q=\mathrm{\infty }$ it follows from the essential bounds for the two summands. Real completeness is [F1], using the Countable Choice assumed in [F4]. For complex completeness, a Cauchy sequence $(f_j)$ has Cauchy real and imaginary parts because each component norm is at most the complex norm. Apply real completeness to obtain limits $u$ and $v$ in real $L^q$. The triangle inequality gives $\mathrm{\parallel }{f}_j-(u+iv){\mathrm{\parallel }}_q\le \mathrm{\parallel }\mathrm{Re}{f}_j-u{\mathrm{\parallel }}_q+\mathrm{\parallel }\mathrm{Im}{f}_j-v{\mathrm{\parallel }}_q\to 0$. Thus the complex quotient is Banach as well.

The lattice property follows directly from monotonicity of the integral or of the essential supremum. For a finite positive-measure piece $A$, Holder applied to $\mathrm{\mid }y\mathrm{\mid }$ and ${\mathbf{1}}_A$ gives ${\int }_A\mathrm{\mid }y\mathrm{\mid }d\mu \le \mu (A{)}^{1-1\mathrm{/}q}\mathrm{\parallel }y{\mathrm{\parallel }}_q$, with $1\mathrm{/}q=0$ at infinity; zero-measure pieces are trivial. For finite $q$, Fatou applied to $\mathrm{\mid }{y}_j{\mathrm{\mid }}^q$ gives the sequential lower-limit property, and Tonelli gives measurability of $z\mapsto \int \mathrm{\mid }G(z,x){\mathrm{\mid }}^{q}d\mu$ for time or time-pair parameter $z$. Minkowski applied to a jointly measurable kernel $K$ with $\int \mathrm{\parallel }K(r,\cdot ){\mathrm{\parallel }}_{q}dr<\mathrm{\infty }$ gives $H(x)=\int \mathrm{\mid }K(r,x)\mathrm{\mid }dr\in L^q(X)$ and $\mathrm{\parallel }H{\mathrm{\parallel }}_q\le \int \mathrm{\parallel }K(r,\cdot ){\mathrm{\parallel }}_{q}dr$. Hence the scalar integral exists absolutely almost everywhere, and $\mathrm{\mid }\int K(r,x)dr\mathrm{\mid }\le H(x)$ proves its integral-triangle inequality.

For $q=\mathrm{\infty }$, write $a(z)={\mathrm{ess\hspace{0.17em}sup}}_x\mathrm{\mid }G(z,x)\mathrm{\mid }$. For every real $c\ge 0$, the set $\{z:a(z)>c\}$ is the union over the exhaustion pieces of $\{z:\mu (\{x\in A_n:\mathrm{\mid }G(z,x)\mathrm{\mid }>c\})>0\}$. Indeed, a positive-measure strict superlevel set is a countable union of the sets $\{\mathrm{\mid }G\mathrm{\mid }\ge c+1\mathrm{/}k\}$, so some such set has positive measure. Tonelli makes the section measures measurable, hence $a$ measurable. If $y_j\to y$ almost everywhere and $L={\mathrm{lim\hspace{0.17em}inf}}_j\mathrm{\parallel }{y}_j{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$, choose a subsequence with norms at most $L+\epsilon$ eventually. The countable union of its exceptional spatial null sets is null, so $\mathrm{\mid }y\mathrm{\mid }\le L+\epsilon$ almost everywhere by taking limits. Letting $\epsilon \downarrow 0$ proves the endpoint lower-limit property; if $L=\mathrm{\infty }$ the inequality is automatic.

For a jointly measurable kernel $K(r,x)$, set $a(r)=\mathrm{\parallel }K(r,\cdot ){\mathrm{\parallel }}_{\mathrm{\infty }}$, measurable by step 1.3. If $\int a(r)dr<\mathrm{\infty }$, then $a$ is finite almost everywhere, and $\mathrm{\mid }K(r,x)\mathrm{\mid }\le a(r)$ holds for almost every $x$ for each such $r$. Tonelli applied to the measurable violation set on the finite-measure spatial pieces shows that this bound holds for almost every $(r,x)$, hence for almost every $x$ and almost every $r$. Thus its scalar integral is absolutely convergent and has absolute value at most $\int a(r)dr$ for almost every $x$. Taking essential suprema proves the endpoint integral-triangle inequality. Tonelli also shows that changing a joint representative on a product-null set changes its spatial section class only for a null set of parameters, so the scalar mixed norm is well defined. All five hypotheses in Scalar Banach-function retarded estimates by Fatou approximation have now been verified. No $L^{\mathrm{\infty }}$-valued Bochner integral or separability assertion entered this proof.