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Let $\Omega$ be a finite set and $(\Omega,d)$ be metric space with $d$ discrete metric.Let $(\Omega^{\infty},d^{\infty})$ be metric space where $\Omega^{\infty}$ is space of sequences taking values in $\Omega$ and $d^{\infty}$ is the discrete metric. What is the way to construct product measure or any other measure on this space?

I have heard that there is way using Caratheodory's extension theorem.Could someone explain me or give me a reference to read about construction of measures on the given space?

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  • $\begingroup$ You need to give more information, otherwise the question is very vague. For instance, if $x \in \Omega^\infty$, then I can consider $\delta_x$, the Dirac measure at $x$. As you see, there are many possible measures, you have to give some further information allowing us to eliminate most of them and focus on what is of interest to you. $\endgroup$ – Alex M. Sep 11 '16 at 11:29
  • $\begingroup$ @AlexM. I don't know about Dirac measure.Do you have any reference to read about measures on arbitrary product spaces? $\endgroup$ – Math Lover Sep 11 '16 at 11:34
  • $\begingroup$ Measures on product spaces are in no way different from measure on any other abstract spaces, unless you also put a measure on $\Omega$ - which you do not do. If you decide to put a measure $P$ on $\Omega$ and want to construct the product measure $P^\infty$ on $\Omega^\infty$, then P must be a probability, otherwise your product measure will be either $\infty$ or $0$. $\endgroup$ – Alex M. Sep 11 '16 at 11:38
  • $\begingroup$ @AlexM: Sorry..why would it be zero or $ \infty$ if we dont put measure on $\Omega$ ? $\endgroup$ – Math Lover Sep 11 '16 at 11:40
  • $\begingroup$ That is not what I have said: it would be $0$ or $\infty$ if you put a measure on $\Omega$ that were not a probability. $\endgroup$ – Alex M. Sep 11 '16 at 11:44

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