I am new to measure theory and here is the definition I have :
(1) A function f:(X,A)→(Y,B) is mesurable iff
: ∀B∈B,f−1(B)∈A
Why this definition and not this one ?
(2) A function f:(X,A)→(Y,B) is mesurable
iff : ∀A∈A,f(A)∈B
Thus with (2) a function is measurable iff it maps measurable sets to measurable sets. It's seems more natural to me.
I know that the first definition extend the notion of continuity, but this explanation still doesn't convince me that (1) should be the most natural definition.
So are functions that respect (2) have a name ? And why (2) is not the definition of measurable functions ?
