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I am trying to wrap my head around the following statement, which involves a monotone transformation of random variables.

Let $n\in\mathbb{N}$ be fix and $\{A_{i}\}_{i=1,\ldots,n}$ a family of non-negative random variables with finite expectation, i.e. $A_{i}\in\mathcal{L}^{1}(\Omega,\mathcal{F},\mathbb{P})$.

The statement comes from a working paper without proof and reads: If $T:[0,\infty)^{n}\rightarrow[0,\infty)$ is a non-decreasing function, then

  1. $T$ is measurable (and $T(A_{1},\ldots,A_{n})$ therefore a "proper" random variable);
  2. The expectation of $T(A_{1},\ldots,A_{n})$ exists and is finite.

As far as claim 1 is concerned, (Borel-)measurability should be added as an additional requirement in the definition of the transformation $T$, don't you think? I believe the answers to the following post support my concerns: Monotone functions are measurable.

Now, for claim 2. I think that integrability should be implied by monotonicity, although I failed to come up with a valid argument/reference. Finiteness of expectation on the other hand seems awefully far fetched - or am I missing something?

Moreover, I don't think that measurability (claim 1) is really needed for integrabiliy (claim 2).

Any comments, ideas or references are highly appreciated.

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    $\begingroup$ I think monotonicity should automatically imply Borel-measurability. Claim 2, however, is nonsense. If $n=1$ and $T(x)=x/(1-x)$, then if $A_1$ is uniformly distributed on the unit interval, $T(A_1)$ has infinite expectation. $\endgroup$
    – Anthony Quas
    Commented Dec 15, 2016 at 18:29

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