Proof of Cumulative-Rate Time Change: Regularity, Substitution, and Crossing Times

Claim 1. The integrand defining $A(t)$ lies between $0$ and $B$, so monotonicity of the integral and $\lambda_{[0,T]}([0,T])=T$ (claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval) give $0\le A(t)\le BT<\infty$ for every $t$. Also $A(0)=\int_{[0,T]}a\,\mathbf{1}_{\{0\}}\,d\lambda_{[0,T]}=0$, since the integrand is bounded by $B\,\mathbf{1}_{\{0\}}$ and $\lambda_{[0,T]}(\{0\})=0$, the singleton being an interval of Lebesgue measure $0$ by claim 4 of Existence of Lebesgue Measure on the Real Line. For $0\le s\le t\le T$, $a\,\mathbf{1}_{[0,t]}-a\,\mathbf{1}_{[0,s]}=a\,\mathbf{1}_{(s,t]}$, which lies between $0$ and $B\,\mathbf{1}_{(s,t]}$; since all quantities are finite, linearity and monotonicity give $0\le A(t)-A(s)\le B\,\lambda_{[0,T]}((s,t])=B(t-s)$. Continuity follows.

Claim 2, measurability. $A$ is continuous, hence $\mathcal{B}_{[0,T]}$-measurable by claim 3 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. For Borel-measurable $f:\mathbb{R}\to[0,\infty]$ the composition $f\circ A$ is $\mathcal{B}_{[0,T]}$-measurable, since $(f\circ A)^{-1}(E)=A^{-1}(f^{-1}(E))$ for every Borel set $E$. For the product with $a$: for real $c<0$ the superlevel set $\{(f\circ A)\,a>c\}$ is $[0,T]$; for $c\ge0$, $\{(f\circ A)\,a>c\}=\bigcup_{q}\Bigl(\{f\circ A>q\}\cap\{a>c/q\}\Bigr),$ the union over all positive rationals $q=z/n$ with $z,n\ge1$ natural numbers, listable as a sequence by increasing $n$ and, within each $n$, increasing $z$; each member of the union lies in $\mathcal{B}_{[0,T]}$. Indeed, if $f(A(t))\,a(t)>c\ge0$ then $a(t)>0$ and $f(A(t))>0$, and by the Archimedean property there is a positive rational $q$ with $c/a(t)<q<f(A(t))$ (taking $q$ arbitrarily large when $f(A(t))=\infty$), so that $f(A(t))>q$ and $a(t)>c/q$; conversely $f(A(t))>q$ and $a(t)>c/q$ give $f(A(t))\,a(t)>q\cdot(c/q)=c$. This proves the measurability assertion of claim 2. For the special case $f=\mathbf{1}_E$ it also follows directly: the product is $\mathbf{1}_{A^{-1}(E)}\,a$, a product of real-valued measurable functions, measurable by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable applied to the sequentially continuous map $(x,y)\mapsto xy$.

Claim 2, the identity. Define, for Borel $E\subseteq\mathbb{R}$, $\mu_1(E)=\int_{[0,T]}\mathbf{1}_E(A(t))\,a(t)\,d\lambda_{[0,T]}(t),\qquad \mu_2(E)=\lambda\bigl(E\cap[0,A(T)]\bigr).$ Both are measures: for $\mu_1$, if $E$ is the disjoint union of $E_1,E_2,\dots$ then $\mathbf{1}_E(A(t))\,a(t)$ is the nondecreasing pointwise limit of the finite partial sums of $\mathbf{1}_{E_n}(A(t))\,a(t)$, and additivity together with the Monotone Convergence Theorem gives countable additivity; $\mu_2$ is the measure with density $\mathbf{1}_{[0,A(T)]}$ with respect to $\lambda$. Both are finite with total mass $A(T)$.

They agree on the rays $(-\infty,u]$. If $u<0$ both sides vanish, since $A\ge0$. If $u\ge0$, let $g(u)$ be the least upper bound of the nonempty set $\{t\in[0,T]:A(t)\le u\}$; by continuity $A(g(u))\le u$, and by monotonicity $\{t\in[0,T]:A(t)\le u\}=[0,g(u)]$. If $g(u)<T$ then $A(s)>u$ for every $s>g(u)$, so by continuity $A(g(u))=u$; in either case $A(g(u))=\min(u,A(T))$. Hence $\mu_1\bigl((-\infty,u]\bigr)=\int_{[0,T]}\mathbf{1}_{[0,g(u)]}(t)\,a(t)\,d\lambda_{[0,T]}(t)=A(g(u))=\min\bigl(u,A(T)\bigr)=\mu_2\bigl((-\infty,u]\bigr).$ By claim 2 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law, the rays $(-\infty,u]$ form a $\pi$-system generating the Borel $\sigma$-algebra, so claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law gives $\mu_1=\mu_2$.

For general measurable $f:\mathbb{R}\to[0,\infty]$, the identity $\int_{[0,T]}\phi(A(t))\,a(t)\,d\lambda_{[0,T]}(t)=\int_{\mathbb{R}}\phi\,\mathbf{1}_{[0,A(T)]}\,d\lambda$ holds for indicators $\phi=\mathbf{1}_E$ by the equality $\mu_1=\mu_2$ and claim 3 of Image Measures, Measures with Densities, and Change of Variables; for simple functions by linearity; and for general $f$ by the Monotone Convergence Theorem applied on both sides along the nondecreasing simple functions $s_n=\sum_{p=1}^{n2^n}2^{-n}\,\mathbf{1}_{\{f\ge p\,2^{-n}\}}$, which converge pointwise to $f$ (at a point where $f$ is finite, $s_n$ is within $2^{-n}$ of $f$ once $n$ exceeds $f$ at that point; at a point where $f$ is infinite, $s_n=n$ there).

Claim 3. For $0<u\le A(T)$ the set $L(u)$ contains $T$ and is closed under limits by continuity, so its greatest lower bound $\kappa(u)$ belongs to it: $A(\kappa(u))\ge u$. Since $A(0)=0<u$, $\kappa(u)>0$; for $0\le t<\kappa(u)$ we have $A(t)<u$, and letting $t$ increase to $\kappa(u)$, continuity gives $A(\kappa(u))\le u$; hence $A(\kappa(u))=u$. Monotonicity now gives $L(u)=\{t\in[0,T]:t\ge\kappa(u)\}$: if $t\ge\kappa(u)$ then $A(t)\ge A(\kappa(u))=u$, and if $t<\kappa(u)$ then $A(t)<u$ by the definition of the greatest lower bound. For $u>A(T)$, monotonicity gives $A(t)\le A(T)<u$ for all $t\in[0,T]$, so $L(u)$ is empty.

Claim 4. If $A(T)=0$ both sides vanish ($a$ has integral $0$, so by claim 5 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval it vanishes off a $\lambda_{[0,T]}$-null set, and the right side is $0$ by splitting the integral over that null set and its complement). Assume $A(T)>0$. By claim 3, for $0<u\le A(T)$ and $s\in[0,T]$: $\kappa(u)\le s$ if and only if $A(s)\ge u$, so $\{u\in(0,A(T)]:\kappa(u)\le s\}=(0,A(s)].$ Measurability of $\kappa$: the sets $[0,s]$ with $s\in[0,T]$ form a $\pi$-system generating the trace Borel $\sigma$-algebra on $[0,T]$ (each $[0,s]$ is the trace of the ray $(-\infty,s]$; the Borel sets whose traces lie in the $\sigma$-algebra generated by the family $\{[0,s]\}$ form a $\sigma$-algebra containing all rays $(-\infty,s]$, whose traces are $[0,s]$, $[0,T]$, or $\emptyset$, hence containing every Borel set by claim 2 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law); and the class of trace Borel sets $E$ with $\kappa^{-1}(E)$ a trace Borel subset of $(0,A(T)]$ is a $\sigma$-algebra containing the sets $[0,s]$, by the display. Hence $\kappa$ and every composition $\Phi\circ\kappa$ are measurable.

Define, for trace Borel $E\subseteq[0,T]$, $\nu_1(E)=\int_{\mathbb{R}}\mathbf{1}_E(\kappa(u))\,\mathbf{1}_{(0,A(T)]}(u)\,d\lambda(u),\qquad \nu_2(E)=\int_{[0,T]}\mathbf{1}_E\,a\,d\lambda_{[0,T]}.$ Both are measures (countable additivity by finite partial sums of indicators, additivity, and the Monotone Convergence Theorem, as in claim 2), both finite with total mass $A(T)$ (for $\nu_1$: $\lambda((0,A(T)])=A(T)$ by claim 4 of Existence of Lebesgue Measure on the Real Line). They agree on the $\pi$-system $\{[0,s]:s\in[0,T]\}$: by the display, $\nu_1([0,s])=\lambda((0,A(s)])=A(s)=\nu_2([0,s])$. Since that $\pi$-system generates the trace Borel $\sigma$-algebra, claim 1 of Uniqueness of Finite Measures on a Generating Pi-System and the Density of the Exponential Law gives $\nu_1=\nu_2$. The identity of claim 4 then holds for indicators, for simple functions by linearity, and for general $\Phi$ by the Monotone Convergence Theorem along the nondecreasing simple approximations used in claim 2.