# An example of a projective sequence of measures on non-Borel spaces that does not extend to a probability on the product space?

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.

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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