Continuous-time linear state-space model

Assume the Axiom of Countable Choice, so the usual Lebesgue-measurable sets on $\mathbb{R}$ form a complete measure space with Lebesgue measure $\lambda$. Every bounded interval is Lebesgue measurable, and any interval is the countable union of its bounded intersections with $[-n,n]$; since the Lebesgue-measurable sets form a sigma-algebra, $I$ is measurable and $\lambda$ restricts to $I$. Let $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$, let $I\subseteq \mathbb{R}$ be an interval with $t_0\in I$, and let $N,m,p\ge 1$ be finite. A continuous-time linear state-space model consists of matrices $A\in {\mathbb{F}}^{N\times N}$, $B\in {\mathbb{F}}^{N\times m}$, $C\in {\mathbb{F}}^{p\times N}$, $D\in {\mathbb{F}}^{p\times m}$, an initial value $x_0\in {\mathbb{F}}^N$, and an input $u:I\to {\mathbb{F}}^m$ whose real and imaginary coordinates are Lebesgue measurable and integrable on every compact subinterval of $I$. A state trajectory is a function $x:I\to {\mathbb{F}}^N$ whose real and imaginary coordinates are absolutely continuous on every compact subinterval of $I$ (with absolute continuity defined by the $\epsilon$–$\delta$ condition on finite families of disjoint subintervals), with $x(t_0)=x_0$ and $x^{\mathrm{\prime }}(t)=Ax(t)+Bu(t)$ outside a Lebesgue-null subset of $I$. Its output is $y(t)=Cx(t)+Du(t)$. The term $Du(t)$ is the direct input-to-output term and is separate from the state contribution $Cx(t)$.

Gu, Goel, and Ré, “Efficiently Modeling Long Sequences with Structured State Spaces,” §2.1

Integrable real and complex functions, and their integrals

Absolute continuity on a compact interval

Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$

Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)