Explicit example of second Borel–Cantelli lemma

Question

Consider the probability model $(\Omega, \mathcal{F}, P)$ where $\Omega = [0,1]$, $\mathcal{F}$ is the Borel $\sigma$-algebra on $[0,1]$ and $P$ is the uniform measure on $[0,1]$.

Let $E_1, E_2, \dots$ be a sequence of independent events in $\mathcal{F}$ such that $P(E_n) = 1/n$.

Since $\sum P(E_n) = \infty$, by the second Borel–Cantelli lemma (Wikipedia), $$ P(\bigcap_{k=1}^\infty \bigcup_{n=k}^\infty E_n) = 1. $$

My question is:

Is there any explicit example of such $E_n \subset [0,1]$?

The most famous example of independent events on $[0,1]$ is the dyadic intervals: Define

$D_1 = [0, 2^{-1}]$

$D_n = 2^{-1}D_{n-1} + (2^{-1}+2^{-1}D_{n-1}) $

$(D_n)_{n =1}^\infty$ is a sequence of independent events. However, for this example, $P(D_n) = 1/2^n$.

Is there any explicit example for $P(E_n) = 1/n$ ?

Answer 1

All irrational numbers in $(0,1)$ have unique representation as $\sum_{k=2}^\infty c_k/k!$, where $c_k\in \{0,1,\dots,k-1\}$. 'Digits' $c_k$ are independent, so you may choose the events $E_n$ as '$c_n=0$', for example.

Answer 2

Construction of such an example essentially amounts to an explicit decomposition of the unit interval $(\Omega,P)$ into a product of probability spaces. For instance, one can take an isomorphism between $(\Omega,P)$ and its infinite countable product by itself $(\Omega,P)\times(\Omega, P)\times \dots$. [Since $(\Omega,P)$ is a countable product of the dyadic space $\{0,1\}$ endowed with the uniform measure by itself, an explicit rearrangement of binary digits will then provide a required isomorphism.] Now, as it was suggested in Alex.B's comment, you just take in each of the multipliers a set of any measure you like.

Answer 3

It is awkward to construct an event of prob 1/17, e.g. out of dyadic, or even decimal digits. As you want a borel cantelli example in the unit interval, why don't you change your probabilities instead to make the point the same but the construction easier. Make e.g. $E_{17}$ a set of probability $[1000/17]/1000 = .058$. Then your event might be decimal digits 49 -51 when placed in that order are a number < 58.