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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?

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  • $\begingroup$ 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? $\endgroup$ – Buschi Sergio 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 $j=1,2$. $\endgroup$ – Pietro Majer Apr 5 '12 at 9:22
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    $\begingroup$ 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. $\endgroup$ – Buschi Sergio Apr 5 '12 at 13:45
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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.

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