Let $(X_m)$ be a sequence of i.i.d. rv's and $n$ a Poisson distributed rv. How is $Y = (X_1, \ldots, X_n)$ indeed a rv? Let $(X_m)$ be a sequence of i.i.d. random variables taking values in the interval $[0, 1]$. Let $n$ be Poisson distributed with parameter $\lambda>0$. Intuitively, $Y = (X_1, \ldots, X_n)$ is a random variable taking values in the space $[0, 1]^{< \omega}$ of all finite $[0, 1]$-valued sequences. To be specific,
$$
[0, 1]^{< \omega} := \bigcup_{i = 1}^\infty [0, 1]^i.
$$
However, I don't know

*

*which $\sigma$-algebra $[0, 1]^{< \omega}$ is endowed with, and

*how $Y$ is indeed a random variable.

Could you elaborate on my confusion?

There is no standard $\sigma-$ algebra on $[0, 1]^{< \omega}$ (as far as I know) but it would be natural to say that $E$ is measurable in $[0, 1]^{< \omega}$ if $E \cap [0,1]^{i}$ is measurable for each $i$. Note that $Y \in [0,1]^{i}$ forces $n$ to be equal to $i$ so measurablity of $Y$ is quite easy to check with this definition.
