conditional probability as the conditional expectation of an indicator

There is a single idea that lets the whole machinery of conditional expectation also handle conditional probability, so you never need a separate theory for events. The trick is the indicator variable 1_A, which equals 1 when event A happens and 0 when it does not. Its plain expectation is just the probability of A: E[1_A] = P(A). Conditioning that same indicator on information G turns it into a conditional probability.

Precisely: P(A given G) is defined as E[ 1_A given G ]. This is a random variable — your best estimate, using the information in G, of the chance that A occurs. All the conditional-expectation properties transfer for free: it is nonnegative and at most 1 (by positivity applied to 0 <= 1_A <= 1), the tower property becomes the law of total probability, and 'taking out what is known' specializes to facts about events you already know. When G is generated by a variable Y, P(A given Y) is the familiar function-of-Y you met in elementary conditioning, now firmly grounded.

This unification is more than tidiness — it is what makes the modern theory uniform. But it hides a genuine subtlety. For each fixed event A, P(A given G) = E[1_A given G] is defined only up to a probability-zero set, and there are uncountably many events A. The exceptional null sets can pile up as A varies, so there is no automatic guarantee that the family { P(A given G) : A } behaves like an honest probability measure (countably additive) for almost every outcome simultaneously. Repairing this — choosing the versions coherently — is exactly the problem of the regular conditional distribution.