which is useful when you define integrals and expectations:
E g(Y) = ∫ g(y) f(y) dy = ∑ᵢ g(cᵢ) P(y ∈ Aᵢ)
where Y is a random variable with density function f. Any integrable
function can be approximated as the limit of step functions, so this
is a well-behaved way to get a general theory of integration.
Of course, one could replace (y ∈ Aᵢ) with 1 - (y ∈ Aᵢ) if one wanted
to use "0" to represent the event (y ∈ Aᵢ) and "1" to represent its
complement and not affect the truth of the math, but then there will
be lots of termf floating around just to convert the notation into the
terms that you need for the math.
g(y) = ∑ᵢ cᵢ × (y ∈ Aᵢ)
which is useful when you define integrals and expectations:
E g(Y) = ∫ g(y) f(y) dy = ∑ᵢ g(cᵢ) P(y ∈ Aᵢ)
where Y is a random variable with density function f. Any integrable function can be approximated as the limit of step functions, so this is a well-behaved way to get a general theory of integration.
Of course, one could replace (y ∈ Aᵢ) with 1 - (y ∈ Aᵢ) if one wanted to use "0" to represent the event (y ∈ Aᵢ) and "1" to represent its complement and not affect the truth of the math, but then there will be lots of termf floating around just to convert the notation into the terms that you need for the math.