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

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The discussion clarifies the relationship between the measurability of functions in the context of Tonelli/Fubini's Theorem. It establishes that stating "y ↦ ∫_E f^y(x) dx is measurable" emphasizes the function's dependence on y, while "∫_E f^y(x) dx is measurable" serves as a more concise expression. Both statements convey the same mathematical truth regarding the measurability of the integral as a function of y, highlighting the importance of notation in mathematical communication.

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

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