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May 15th, 2019, 01:32 AM   #1
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measure of

It is a standard thing in ergodic theory to determine when the following is true,

$$\lim\limits_{k \to \infty} \dfrac{1}{k} \sum_{k=1}^{n} I_{A}(f^k(x)) = \mu(A)$$

where $I_A$ is an indicator function, $(X,f)$ a dynamical system with $A \subset X$. But I don't understand how we are able to deduce the size of an uncountable set $A$ with a countable operation (the sum). Never mind taking limits to infinity or saying that $x$ explores densely the set $A$, it's not enumerable.

What am I missing..? I think the simplest case would be to understand the proof for the case of an irrational rotation $f_c(x) = x + c \ \text{mod} \ 1$ but I couldn't find a decent resource. Links appreciated!
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May 16th, 2019, 10:43 PM   #2
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Thought I would've been mauled by now
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May 16th, 2019, 11:58 PM   #3
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There's not much to respond. The result you mention is true. It's just not intuitive. I agree with that, it is not intuitive to me too. But then intuition needs to be updated.
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May 17th, 2019, 01:51 AM   #4
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I just realised there is an error with the indices in the OP, but should be clear what was meant..
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