Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Daniell proved a theorem on the existence of random sequences (see page 13 of these notes):

Let $(S_n,\mathbf{S_n})$ be a sequence of Borel spaces and let $\mu_n$ be a projective sequence of probability measures on $ (S_n:n\in > \mathbb{N})$. Then, there's a unique probability measure $ \mathbb{P}$ on the product $ \sigma$-algebra of $ \prod_{n\geq 0}S_n$, such that, for all $ n$, for all $ E \in S_0\otimes \cdots\otimes S_n$, $\mathbb{P}[E\times \prod_{i\geq n+1}S_i] = \mathbb{P}[E]$

I'm trying to find a counterexample when we don't assume that $(S_n,\mathbf{S_n})$ are Borel.

Perhaps the simplest examples of non-Borel spaces are finite measurable spaces for which not all singleton sets are measurable.

Non-Borel sets are also non-Borel spaces, but this seems like a complicated way to go. (Quick, relevant question: are all probability measures on such spaces inner regular?)

The main idea of the proof of this theorem seems to be that we can approximate measurable sets on $([0,1],\mathcal{B}[0,1])$ by compact sets, and use the fact that $(S_n:n\in \mathbb{N})$ have the same structure, since they are Borel.

I still don't have a clear idea of which properties any counterexample must satisfy, which is probably why I haven't got very far!

Thank you.

share|improve this question
See H. Wegner, On consistency of probability measures, Z. Wahrscheinlichkeitstheorie verw. Geb. 27, Nr. 4, (1973), 335-338, and the references therein. –  t.b. Jul 17 '11 at 15:44
@theo: Thank you. The paper refers to the book Measure Theory by Halmos, which has the required example on page 214. –  Ben Derrett Jul 17 '11 at 17:08
add comment

1 Answer

The historically first such example was in a paper by Andersen and Jessen, On the introduction of measures in infinite product sets. Danske Vid. Selsk., Mat.-Fys. Medd. 25, No.4, 7 S. (1948).

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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