The A=a notation always bugged me too.
I like the above notation because it betrays morphism composition.
If we consider random variables as measure(able) spaces and conditional probabilities P(B | A) as stochastic maps B -> P(A), then every element 'a' of (a countably generated) A induces a point measure -> A giving probability 1 to that event. This is the map named by do(a). But since we're composing maps, not elements, we can use an element a unambiguously to mean its point measure. Then a series of measures separated by ',' give the product measu... (read more)
The A=a notation always bugged me too. I like the above notation because it betrays morphism composition.
If we consider random variables as measure(able) spaces and conditional probabilities P(B | A) as stochastic maps B -> P(A), then every element 'a' of (a countably generated) A induces a point measure -> A giving probability 1 to that event. This is the map named by do(a). But since we're composing maps, not elements, we can use an element a unambiguously to mean its point measure. Then a series of measures separated by ',' give the product measu... (read more)