I'm reading a text "Lebesgue Integration - Frank jones" from which i got recommended here, stackexchage.
This text seemingly covers various topics on measure theory, but i think that's it. This text seems student-friendly at the first glance, but the more i look careful, the more i found it very terse.
It's not even this text, but other texts don't really care about measurability.
Here's a phrase in a text
"Frank Jones - Lebesgue Integration - p.203"
Let $a_1,\cdots,a_n$ be positive real numbers.
Let $f$ be a Lebesgue measurable function on $(0,\infty)$
Let $ L =\int_{\mathbb{R}_+^{n+1}} {x_1}^{a_1-1}\cdots{x_{n-1}}^{a_{n-1}-1}dx_1\cdots dx_n \int_{x_1+\cdots+x_{n-1}}^\infty f(t)(t-x_1-\cdots-x_{n-1})^{a_n -1} dt$
Then by Tonelli's theorem, $L =\int_0^\infty f(t) dt \int_{x_1+\cdots x_{n-1}<t \bigwedge x_k>0} {x_1}^{a_1-1}\cdots{x_{n-1}}^{a_{n-1}-1}(t-x_1-\cdots-x_{n-1})^{a_n -1} dx_1\cdots dx_n$
I don't see any logic from second line to third line in the above phrase.
In order to apply Tonelli's theorem, it must be checked that $H((x_1,\cdots,x_{n-1}),y) = f(y){x_1}^{a_1-1}\cdots{x_{n-1}}^{a_{n-1}-1}(t-x_1-\cdots-x_{n-1})^{a_n -1}$ is Lebesgue measurable on its natural domain
And i have no idea how to prove that this is measurable and posted a question about this and i haven't got any answer there. ( How do i prove this mixture of Lebesgue measurable functions is Lebesgue measurable?)
Why do people just neglect such hypothesis in Fubini's Theorem?