Suppose that is a measure space, a measurable set with . We can create a new measure space where

Note that is a probability space, as , and any other set is a subset of .

Supposing is a probability space, we can use this new probability space in our definition of conditional probability. The probability represents the probability of occurring, where we already know has occurred. Normally, instead of going through the trouble of writing out a new sigma-algebra and probability measure each time, we simply take to be the probability of using the measure defined above. Of course, our measure and sigma-algebra are so simple that we can just write this in one line as

We call this the probability of given . Now if , we say that and are independent events. If this is the case then we have

This is certainly a useful property as it makes proofs of interesting facts fall out easily when we consider sequences of independent random variables (a related idea) later.

Now consider the fact that implies . Plugging into the formula for we arrive at the following, known as Bayes’ theorem.

Note that .

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