I want to identify the interval $[0, 1]$ with the Lebesgue measure to the probability space for infinite tossing a fair coin.
I know we can define a probability on cylinder sets (the sets can be represented by the outcomes of finite tosses) and then use Caratheodory extension theorem to extend the probability to the $\sigma$-algebra $\mathcal{F}$ generated by all the cylinder sets.
But I am not sure how to identify this probability measure with the Lebesuge measure on $[0,1]$. In particular, I don't know what $\mathcal{F}$ looks like and why it can be identified as $\sigma$-algebra where Lebesgue measure is defined. I just have a brief thought about identify $\omega\in[0,1]$ with a binary representation.
Could you explain how to identify these two measure space ? It will also be helpful if you can provide a reference about this topic.