0
$\begingroup$

Let

  • $n\in\mathbb{N}$,
  • $I_j\subseteq\mathbb{R}$, $1\le j\le n$, intervals,
  • $I:= I_1\times\dots\times I_n\subseteq\mathbb{R}^n$,
  • $f_j:I_j\to\mathbb{R}$, $1\le j\le n$, Lebesgue measurable,
  • $g:I\to\mathbb{R}$ continuous,
  • $h:I\ni(x_1,\dots,x_n)\mapsto g(f_1(x_1),\dots,f_n(x_n))\in\mathbb{R}$.

Then $h$ is Lebesgue measurable.

Do you know a textbook reference of this statement?

  • 0
    For reference: http://math.stackexchange.com/questions/97182/cauchy-formula-for-repeated-lebesgue-integration2012-01-07

1 Answers 1

2

The $n=2$ case is essentially covered by Theorems 1.7 and 1.8 in Rudin's Real and Complex Analysis.

It's not exactly what you want--- he assumes the domains of the $f_i$ are all equal to the same measurable space--- but this isn't essential to the proof, and once you see the proof you will understand how to modify it accordingly.

And of course the idea for general $n$ is the same (and exactly that which was outlined in t.b.'s comment; what he alludes to about left composition and measurability is a generalization of Rudin's Theorem 1.7).

  • 0
    @precarious: Yes.2012-01-08