What is measure of numbers with certain property on [0,1]

[dimitri151]

Considering the interval [0,1], say for each number (binary) on the interval you form the sequence of numbers: number of zeros up to the nth place/number of ones up to the nth place. Then as n goes to infinity the sequence of numbers (for the given binary number) will go to somewhere in [-infinity,infinity] if they converge. What is the measure of numbers on [0,1] that have this sequence converge vs not converge, and of those that converge what are the measures with numbers that converge to <1, 1 and >1?

 

[mfb]

Edit: Ah, I think I understand your sequence now.
The nth term is the ratio of "0" to "1" in the first n terms of the binary expansion?

All sequences for rational numbers will certainly converge to some rational number. "Most" sequences for irrational numbers should converge to 1.

 

[lavinia]

I think that the function $f_{n}(x)$ that takes the ratio of zeros to ones in the first n places is Lebesque measurable. For instance, $f_{1}$ assigns 0 to the numbers less than 1/2 and 1 to the numbers greater than or equal to 1/2.

I think that these functions converge pointwise to 1/2 almost everywhere.

Would like to see a proof. Do you have any ideas?

 

[Hawkeye18]

It is just the law of large numbers that the functions $f_n$ converge to $1$. Namely, consider random variables $X_n$, $X_n(\omega)=$$n$th digit of $\omega\in[0.1]$. Then the random variables $X_n$ are independent (easy to check by definition), identically distributed with mean value one (and with bounded variance), so $$\frac{1}{n}\sum_{k=1}^n X_n \to \frac12;$$ convergence here is in probability (i.e. in measure) by the weak law of the large numbers, and almost sure (almost everywhere) by the strong law.

Convergence in probability implies that for almost all $\omega\in[0,1]$ the ration of 0s and 1s has limit 1. And the strong law implies that for $f_n$ from lavina's post $f_n\to 1$ a.e.

 

[lavinia]

It is just the law of large numbers that the functions $f_n$ converge to $1$. Namely, consider random variables $X_n$, $X_n(\omega)=$$n$th digit of $\omega\in[0.1]$. Then the random variables $X_n$ are independent (easy to check by definition), identically distributed with mean value one (and with bounded variance), so $$\frac{1}{n}\sum_{k=1}^n X_n \to \frac12;$$ convergence here is in probability (i.e. in measure) by the weak law of the large numbers, and almost sure (almost everywhere) by the strong law.

Convergence in probability implies that for almost all $\omega\in[0,1]$ the ration of 0s and 1s has limit 1. And the strong law implies that for $f_n$ from lavina's post $f_n\to 1$ a.e.

Right - it is the Strong Law of Large Numbers.