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Looking at the proof of the Law of large numbers, you can tell it doesn't refer directly to the definition of expected value.

I know it's wrong to assume the LLN would hold with a different definition of $E[X]$, but my question is which part of the proof exactly would be invalid if we chose a different definition.

In the proof above nothing would change. So one of the following would have to get broken in that case: Markov's inequality or Chebyshev inequality which uses Markov's inequaliy. Could anyone tell which theorem wouldn't hold with a different definition of expectation?

The LLN proves that long time average converges to what's currently defined as expected value. If we change the definition of expected value, it will simply be a different number, so LLN can't hold with any different definition of $E[X]$. If a series converges to, say, $n$, it cant' converge to $2n$ or $3.14$ or anything else at the same time. But my question is which part of the proof of LLN would get broken (I know something has to break there).

user216094
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    It is impossible to answer this question unless one knows what alternative definition is under consideration. – Michael Oct 05 '15 at 21:26
  • Well, people say the current definition of expectation is right because it makes LLN true (it means no other definitions would work). – user216094 Oct 05 '15 at 21:55
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    Well, the proof uses the facts that expectation is linear, monotone and that $P{X \in B}=E(1_{{X \in B}})$ – Zoran Loncarevic Oct 06 '15 at 12:59

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