I don't understand, because
and the function you write is
In this case,
would just be the function that takes every x and add y to it.
It is just a notation.
is the function
, or if you prefer the lambda-calculus notation,
. Of course, this notation doesn't tell you the domain or the codomain of the function, which needed to be specified elsewhere, e.g.
--- it is the function that takes whatever x you input, and spits out the value
, i.e. the function that takes x in X as input, and output the integral of the nonnegative (measurable) function
with respect to measure
on Y. Note the two different types of arrows here ---
is used for specifying the domain/codomain, and
is used for specifying the value of the function at (generic) argument.