It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary.


I was trying to explore what happens if you follow the same procedure with a biased coin. I managed to prove that if the induced measure is absolutely continuous with respect to the Lebesgue measure, then the density must be discontinuous on at least a dense set (which includes all rational numbers in [0, 1] with a finite binary expansion).


But I suspect it's much worse than that. When I try to visualize the cumulative distribution function, it seems to have a "fractal staircase" type of shape, so I suspect the measure is, in fact, singular, but I can't prove it.


Q1. Are there any standard techniques for proving a measure is singular ?


Q2. I strongly suspect that a natural question like this has already been resolved. Any link to a textbook and/or paper would be appreciated.



Measure induced on [0, 1] by infinite tosses of biased coin
For $\omega \in [0,1]$, let $X\_i(\omega)$ be the $i$th binary digit of $\omega$. (If $\omega$ is a dyadic rational and thus has two binary expansions, let's say we choose the expansion that ends with all 0s; it makes no difference). Under Lebesgue measure, the $X\_i$ are iid Bernoulli $1/2$ random variables, so by the strong law of large numbers, $\frac{1}{n}(X\_1 + \dots + X\_n) \to 1/2$ almost surely. In other words, the set $A\_{1/2} := \{\omega : \lim\_{n \to \infty} \frac{1}{n} (X\_1(\omega) + \dots + X\_n(\omega)) = \frac{1}{2}\}$ has Lebesgue measure $1$.


If $\mu\_p$ is the measure coming from a coin that comes up heads with probability $p \ne 1/2$, then under $\mu\_p$, the $X\_i$ are iid Bernoulli $p$. By the strong law again, we have $\mu\_p(A\_p) = 1$. But $A\_p$ and $A\_{1/2}$ are clearly disjoint, so Lebesgue measure and $\mu\_p$ are mutually singular.


I guess the "standard technique" here is to use a zero-one type theorem from probability giving an almost sure statement about limiting behavior, and look for a place where the two measures exhibit different limiting behavior.


