Had a small question: Let's consider the probability space $(\Omega, \mathfrak{F})=([0,1], \mathfrak{B})$ with Lebesgue measure $\mathbb{P}$, $\mathfrak{B}$ is Borel sigma algebra.
Lets expand a point $\omega\in \Omega$ in binary form: $\omega=0.\omega_1\omega_2\omega_3\omega_4\ldots$ and consider a random variable $Y(\omega)$:
$ Y(\omega)=0.\omega_1\omega_3\omega_6\omega_{10}\omega_{15}\ldots $
i.e. the gap between neighbour indeces increase by 1 with each step. It is asserted that this random variable $Y(\omega)$ is uniformly distributed on $\Omega$. Can't figure out why. Would be highly appreciated for any hint.
This paper has an explanation about the uniformity of infinite coin tosses.
Also, there is a detailed explanation given in Santosh Venkatesh's Probability Theory: Explorations and Applications. (starting at p. 144...I linked to the Google preview).
I will not reproduce them here, as these links are excellent.