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it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed. I argue that in the case of a discrete distribution, say a single die, the first two axioms are sufficient as this distribution has a finite number of events. Is this right?

And then I am stretching it a bit by arguing that for simple/well behaved continuous distributions, such as the uniform distribution, the first two axioms are sufficient to ensure a proper probability distribution. Is that right?

I always had the idea that the 3rd axiom was rather there to ensure that more cumbersome distributions or convolutions of distributions would still be proper probability distributions (although I don't have an example at hand). Or is axiom 3 much more vital than I understand?

Thanks for any input! Best, Stefan

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I don't understand what "for simple/well behaved continuous distributions... the first two axioms are sufficient to ensure a proper probability distribution" means. Surely "satisfies sigma-additivity" is a necessary condition for being simple and/or well-behaved. –  Qiaochu Yuan Jan 4 '11 at 14:37
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Note that discrete distributions need not be finite. Perhaps your point is that for a finite event space, finite additivity will suffice. –  hardmath Jan 4 '11 at 14:40
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4 Answers

I am suspecting that you are really asking whether $\sigma$-additivity is really necessary when simple additivity might be enough. If we exclude third axiom as it is defined in wikipedia, then the probability need not to be additive, which is definitely not something you want. Now for finite sample space $\sigma$-additivity and simple additivity coincides, but for infinite sample space $\sigma$-additivity is necessary as other posters pointed out. Take for example probability space $(\Omega, 2^\Omega, P)$ where $\Omega=\{\omega_1,...,\omega_n,...\}$. Now

$$\Omega=\bigcup_{i=1}^\infty\{\omega_i\}$$

and

$$\{\omega_i\}\cap\{\omega_j\}=\varnothing,i\neq j$$

Now $\sigma$-additivity implies

$$P(\Omega)=\sum_{i=1}^\infty P(\omega_i)=1$$

without $\sigma$-additivity this statement need not be true, contradicting axiom 2.

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The power of the third axiom is illustrated here.

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You can make a link directly to a section like this: en.wikipedia.org/wiki/Probability_axioms#Proofs –  Hans Lundmark Jan 4 '11 at 14:37
    
Thanks! Done... –  Shai Covo Jan 4 '11 at 14:44
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The first two axioms given on Wikipedia don't imply finite additivity, which you seem to be assuming. This assumption is enough to do probability with a finite sample space but not with a countable sample space; you need the assumption of $\sigma$-additivity here or else you can't do simple intuitive things like compute the expected number of times you have to roll a die to get a certain result.

If you only assume finite additivity, you get a lot of weird measures; for example any non-principal ultrafilter on an infinite set defines a finitely additive measure. Whether you want to allow these as probability measures is up to you, but the fact is that it's harder to prove theorems in this generality.

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Without the third axiom we can define many distributions that don't work like probabilities are supposed to. For example, the distribution where we assign 1 to all subsets of the universe. That doesn't look like a probability, but it satisfies the first two axioms, so we need an axiom to rule it out.

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