Let $\mathcal{A_1}$ and $\mathcal{A_2}$ be $\sigma$-algebras of subsets of some space X. Suppose $\mu_j$ is probabilistic measure on $\mathcal{A}_j$ for $j=1,2$.

Question: What are the necessary and sufficient conditions for existence of common extension of these measures to some probabilistic measure on $\sigma(\mathcal{A}_1,\mathcal{A}_2)$?

The obvious necessary condition is as follows: $\forall U_i\in \mathcal{A_i}$, $i=1,2$, if $U_1 \subset U_2$ then $\mu_1(U_1)\leq \mu_2(U_2)$ and vice versa. It is known that if we are interested in finitely-additive measures then this condition is sufficient.

What about general $\sigma-$additive measures?

  • $\begingroup$ you wrote: "It is known that if we are interesting in finitely-additive measures then this condition is sufficient" could you give a reference about? $\endgroup$ Apr 5 '12 at 7:54
  • $\begingroup$ @paxa239, by extension you mean without renormalization of the measure? just something like "common refinement"? $\endgroup$
    – Asaf
    Apr 5 '12 at 8:04
  • $\begingroup$ Extension: in the usual sense of functions: a probability measure $\mu$ defined on the larger domain $\sigma(\mathcal{A} _ 1, \mathcal{A} _ 2)$ such that $\mu _{|\mathcal{A} _ j }=\mu _ j$ for $j=1,2$. $\endgroup$ Apr 5 '12 at 9:22
  • 1
    $\begingroup$ Buschi Sergio, a found it in the book K. P. S. Bhaskara Rao, M. Bhaskara Rao "Theory of charges: a study of finitely additive measures", theorem 3.6.1 p.82 $\endgroup$
    – user17150
    Apr 5 '12 at 11:06
  • $\begingroup$ Thank you. I try to study this problem , a key is a characterization of the elements of $\sigma(\mathcal{A}_1,\mathcal{A}_2)$ in terms of those of $\mathcal{A}_1,\mathcal{A}_2$. But I haven't find any useful things about. $\endgroup$ Apr 5 '12 at 13:45

The following example (457H, in Fremlin's Measure theory Vol. 4) shows that the obvious necessary condition isn't sufficient.

Let $X = \{(x, y) \in [0, 1]^2: x < y\}$. Let $\pi_1, \pi_2 : X \to [0, 1]$ be projections on the $x$ and $y$ coordinates. Let $\Sigma_i = \{\pi_i^{-1}[E] : E \subseteq [0, 1] \text{ is Borel}\}$ and $\mu_i(\pi_i^{-1}[E]) = \mu(E)$ where $\mu$ is Lebesgue measure. Then it is easy to check that, although $\mu_1, \mu_2$ have a common finitely additive extension, no such extension can be countably additive.

To appreciate the difficulty in obtaining a simple necessary and sufficient criterion, Fremlin mentions the following concrete problem (457Z(a)): Characterize the Borel sets $X \subseteq [0, 1]^2$ for which the analogously defined measures in the above example admit a common countably additive extension.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.