Is Notation Impacting Measurability in Tonelli/Fubini's Theorem?

[nateHI]

The context is Tonelli/Fubini.

What's the difference between saying
$y\mapsto\int_Ef^y(x)dx$ is measurable
and
$\int_Ef^y(x)dx$ is measurable

 

[mathman]

There really isn't any. All the first statement is doing is emphasizing the fact that the integral is a measurable function of y.

 

[Fredrik]

The second statement is a lazy version of the first. It's like saying that $x^2$ is differentiable instead of saying that $x\mapsto x^2$ is differentiable.

 

[nateHI]

OK thanks!