# Sets and length.

I have a set P in R constructed as follows:

1. Let $E_0 = [0,1]=$ {$.d_1d_2... : 0\leq d_j \leq 9$ for all $j$}.

2. Let $E_1 =${$x \in E_0 : d_1 \ne 0$}

3. Let $E_2 =${$x \in E_1 : d_2 \ne 0$}

4. Continue in this way and define $E_0,E_1,...,E_n$ so that $E_n =${$x \in E_{n-1}: d_n \ne 0$}

5. Define the set P =$\bigcap_{j=0}^{\infty} E_j$

What is the length of P?

I don't know if I could numerate the element of P. I tried numerating then so I got something like this, which I don't think whether is correct.

$E_0 = 010101...$

$E_1 = 101010...$

$E_2 = 110101...$

Now I'm not sure about what exactly the length is.

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Three points:

1. Some numbers have two decimal representations, such as $\frac{1}{2}$ which is $0.5000\ldots$ and $0.4999\ldots$, and these need to be dealt with somehow. But there are a countably infinite number of these compared with the uncountably infinite number of other numbers we are considering, so I will ignore these for now.

2. There are an uncountably infinite number of numbers in $P$: all those with no $0$ in their decimal representation. One way of seeing them is to write the numbers in $[0,1]$ in base $9$ in a similar way; read these as if in base $10$; and add $1$ to each digit.

3. The measure of $P$ is less than or equal to the measure of each $E_n$, which is $0.9$ times the measure of $E_{n-1}$ i.e. $10^{-n}$, so the measure of $P$ must be $0$.

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No, you certainly can’t enumerate the elements of any $E_n$: they’re all uncountable. You’d do much better to try to figure out what these sets actually look like.

$E_0=[0,1]$ is easy: it’s $[0,1]$.

$E_1=\{0.d_1d_2\ldots\in E_0:d_1\ne 0\}$, so the smallest member of $E_1$ is $0.1000\ldots=0.1$, and you should have no trouble verifying that $E_1=[0.1,1]$.

$E_2=\{0.d_1d_2\ldots\in E_1:d_2\ne 0\}$; what elements of $E_1$ does that exclude? Clearly it excludes anything of the form $0.10d_3d_4\ldots$, i.e., everything in the interval $[0.1,0.10\overline{9}]=[0.1,0.11]$. But be careful: it does not exclude $0.11$, which happens to be equal to $0.10\overline{9}$, so it actually excludes only the half-closed interval $[0.1,0.11)$. It also excludes several other intervals of the same length; how many, and what are they? What is their total length? Given that the total length of $E_1$ is $0.9$, what is the total remaining length of $E_2$?

Continue this analysis until you can state and prove a general fact about the total length of $E_n$; then use the fact that $E_0\supseteq E_1\supseteq E_2\supseteq\ldots$ to calculate the total length of $\bigcap_{n\in\Bbb N}E_n$.

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