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Is an integral over an infinite dimensional space defined somewhere? For example, does it make sense to think about

$$\lim_{n \to \infty} \int_{\mathbb{R}^n}f_n(\mathbf{x})\,d\mathbf{x}, \quad \mathbf{x} \in \mathbb{R}^n, \quad f_n:\mathbb{R}^n \to \mathbb{R}$$

or is there something similar to this idea?

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    $\begingroup$ Maybe you meant to have $f$ depend on $n$ as well, i.e. have functions $f_n$, each defined on $\mathbb R^n$. $\endgroup$ – mickep Oct 11 '17 at 6:43
  • $\begingroup$ You are right, thanks. I'll update the question. $\endgroup$ – Student Oct 11 '17 at 6:44
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Since no one else responded, I'll answer with one of my observations.

One scenario where this expression pops up is in applying a limit to an expected value of a (function of) a sequence of random variables.

https://en.wikipedia.org/wiki/Expected_value#Taking_limits_under_the_.7F.27.22.60UNIQ--postMath-000001A0-QINU.60.22.27.7F_sign

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