# Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each $\omega\in\Omega$ and $t\ge 0$ the integral $\int_0^t X_s(\omega)ds$ exists (Lebesgue integration). Why is this integral $\mathcal{F}$-$\mathfrak{B}(\mathbb{R})$-measurable (as a function in $\omega$) if the process $X$ is progressively measurable? Does someone know how to prove this?

Thank you!

• Have a look at this question: math.stackexchange.com/q/1278845
– saz
May 24, 2015 at 16:44
• @saz: Unfortunately, my process $X$ needs not to be continuous whence the Riemann approach could fail to work. I have also tried to use the construction of the Lebesgue integral via simple functions but since I don't know an explicit representation of these approximants I wasn't able to conclude measurability of the integral, yet. May 24, 2015 at 16:57
• Can we show that $(s,\omega) \mapsto X_s(\omega)$ is measurable from $\mathfrak B \times \mathcal F$ to $\mathfrak B$ ?? Then we could probably apply Fubini's theorem. I guess we would need hypothesis $\mathbb E \int_0^t |X_s(\omega)|\,ds < +\infty$ for this approach? May 24, 2015 at 17:18
• @JohnSmith It is not of importance whether the process is continuous; the argumentation in the linked question works for any progressively measurable process such that $\omega \mapsto \int_0^t X_s(\omega) \, ds$ is well-defined.
– saz
May 24, 2015 at 17:26
• Ok, thank you very much! May 24, 2015 at 20:15

example
Without assuming some "joint" measurability of $X_t(\omega)$ in $t$ and $\omega$ you are out of luck.

We use the Continuum Hypothesis. There is a set $A \subseteq [0,1] \times [0,1]$ such that:

$\qquad$For each $t \in [0,1]$,$\qquad \{\omega \in [0,1] : (t,\omega) \in A\}$ is countable,

$\qquad$For each $\omega \in [0,1]$,$\qquad \{t \in [0,1] : (t,\omega) \in A\}$ is co-countable.

On the space $[0,\omega_1)$ of countable ordinals, the set $\{(s,t) : s > t\}$ has these properties; transfer it to $\mathbb R$ by CH.

Now let $\Omega = [0,1]$, $\mathcal F = \mathfrak B$, the Borel sets in $[0,1]$. Also let $\mathcal F_t = \mathcal F$ for all $t \in [0,1]$.

Let $\mathbb P$ be Lebesgue measure.

Choose a non-measurable function $h : [0,1] \to [0,1]$.

Define: $$X_t(\omega) := \begin{cases} h(\omega),\qquad (t,\omega) \in A \\ 0,\qquad (t,\omega) \not\in A \end{cases}$$ Now for each fixed $t$, we have $X_t(\omega)=0$ except for countably many $\omega$, so $X_t$ is $\mathcal F_t$-measurable. [I hope this is your definiton of "adapted $\mathbb R$-valued stochastic process".]

On the other hand, for each fixed $\omega$, we have $X_t(\omega)=h(\omega)$ except for countably many $t$. Integrating a function that is constant a.e., we get $$\int_0^1 X_t(\omega)\;dt = h(\omega)$$ which is a non-measurable function of $\omega$.