In this post, we construct a probability measure on , where denotes the Borel -algebra on the unit interval, which is neither singular nor absolutely continuous. The example comes from Khoshnevisan’s textbook *Probability* published by AMS.

Let be a countable collection of i.i.d. Bernoulli trials with parameter on a probability space . Define a random variable , and let denote the distribution of , so that is a probability measure on .

Our goal is to show that is not absolutely continuous, is nonsingular, and is not a simple combination of the two classes of measures. We will show that for all and that there exists a Borel measurable subset that has zero Lebesgue measure, but .

*Proof*. For , define , and let denote the image of . Note that . Since, for $latex n \in \mathbb{Z}^{\geq 1}$,

is almost surely within of . If, for , we define

then . Since is a decreasing sequence of sets, , where .

We now show that has Lebesgue measure zero. Denote the Lebesgue measure by and observe that, for

Letting , we see that .

Lastly, we show that for all . We can express each in base- as , where . Note that, for any ,

where we use the independence of the to obtain the penultimate expression. Letting , we conclude that .

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