A theorem of Doob. Let (left(mu_{t}ight)_{t geqslant 0}) and (left(u_{t}ight)_{t geqslant 0}) be two families of measures

Question:

A theorem of Doob. Let \(\left(\mu_{t}ight)_{t \geqslant 0}\) and \(\left(u_{t}ight)_{t \geqslant 0}\) be two families of measures on the \(\sigma\)-finite measure space \((X, \mathscr{A})\) such that \(u_{t} \ll \mu_{t}\) for all \(t \geqslant 0\) and \(t \mapsto \mu_{t}(A), u_{t}(A)\) are measurable for all \(A \in \mathscr{A}\). Then there exists a measurable function \((t, x) \mapsto p(t, x),(t, x) \in[0, \infty) \times X\), such that \(u_{t}=p(t, \cdot) \mu_{t}\) for all \(t \geqslant 0\).

[ argue as in the proof of Theorem 25.2 :

\[p_{\alpha}(t, x):=\sum_{A \in \alpha} \frac{u_{t}(A)}{\mu_{t}(A)} \mathbb{1}_{A}(x)\]

and check that this function is jointly measurable in \(t\) and \(x\).]

Data from theorem 25.2

Theorem 25.2 (Radon-Nikodm) Let u, v be two measures on the measurable space (X, A). If  is o-finite, then

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For  book-img-for-question
Question Posted: