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Im using Billingsley's Probability and Measure. In lecture the instructor motivated the need for measure theory in probability by providing a solution to the following problem:

Show that a discrete probability space cannot contain an infinite sequence $A_1, A_2,...$ of independent events, each of probability $\frac{1}{2}$. This is exercise 1.1a in the text.

My thought is that such a sequence may be interpreted as a dyadic expansion of some real number in the unit interval, thus the probability space must be unaccountably infinite. Of course this is not rigorous and it differs from the approach suggested by the author. The problem notes give the following hint:

"Each point lies in one of the four sets $A_1 \cap A_2$, $A_1^c \cap A_2$, $A_1 \cap A_2^c$, $A_1^c \cap A_2^c$ and hence would have probability at most $2^{-2}$; continue."

I'm not sure I see where he is going with this. Of course as the sequence goes infinite, the probability of any set goes to zero, but how does this prove the space cannot be discrete?

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up vote 4 down vote accepted

Let $x$ be an element of the probability space $X$. Let $p$ be the probability of the event $\{x\}$. Consider the $2^n$ pairwise disjoint sets of the form $A_1^{\varepsilon_1}\cap\dots\cap A_n^{\varepsilon_n}$ where for each $i\in\{1,\dots,n\}$, $\varepsilon_i\in\{0,1\}$, $A_i^0=A$ and $A^1_i=X\setminus A$. Each of these sets has probability $2^{-n}$ and $x$ is an element of one of them. It follows that $p\leq 2^{-n}$. Since this holds for every $n$, $p=0$. So every singleton in your space has probability $0$ and hence the space is not discrete.

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