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Let X be a random variable taking the value 0.2 with probability 0.2, 0.4 with probability 0.4 and 0.8 with probability 0.2 and 1.0 with probability 0.2.

Using Infinitary logic I can ask the probability:

$$P(X = P(X = P(X = \ldots)))$$

How is is this value computed?

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I don't see how you can express anything close to anything looking like the thing you wrote with infinitary logic. Maybe you just want to study the set $\{x \in [0;1] / P(X=x) = x\}$ ? – mercio Nov 22 '11 at 13:00
    
If you think this expression belongs to the framework of infinitary logic, you should explain why. Otherwise one could get the feeling that, to you, infinitary logic is a shorthand for everything seemingly paradoxical and with $\cdots$ in it. – Did Nov 27 '11 at 11:21

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