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Suppose I choose a real number between $0$ and $1$.

I can choose its decimal expansion by repeatedly drawing numbers from $D = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, possibly a countably infinite number of times.

But I can also choose its binary expansion by repeatedly drawing numbers from $B = \{0, 1\}$, possibly a countable infinite number of times.

Do I have a way to define a uniform probability measure for both processes that yield the same probability to draw any given real number?

I'm being cautious here because I know uniform probability is not invariant under change of parameter.


Edit: more generally, if I can write the same space as two different Cartesian products, do I have a measure on both that represents the same measure on the orignal space?

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If you take the uniform distribution for digits in each case, you get the uniform distribution on $[0,1]$. To prove this, you check that the measure on sets of the form "the first $n$ digits/bits look like $a_1, a_2, \ldots, a_n$" coincide with the uniform distribution.

If you want to be careful, you can invoke something like Hahn-Kolmogorov for $\sigma$-finite measures to prove uniqueness.

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