Sigma algebra of a random number of random variables

Question

Consider a sequence $\{X_i\}$ of i.i.d. real r.v.'s, and an independent random variable $N$ valued in $\mathbb{N}$, interpreted as a random number of random variables from the sequence, that is $\{X_1,\ldots,X_N\}$. Can we define meaningfully the sigma algebra $\sigma(X_1,\ldots,X_N)$ (I am aware that the notation is meaningless, due to $N$ being random). Conceptually, my guess would be something like $\cap_{i=1}^\infty (\{X_1\in B_1,\ldots,X_i\in B_i, B_k\in\mathcal{B}(\mathbb{R}), 1\leq k\leq i\}\cap\{N=i\})$. However intersecting with the event $\{N=i\}$ ruins the property of this being a sigma algebra. Is there even a way of defining such an object? Thanks for any suggestion. EDIT: I wonder if interpreting $N$ as an independent stopping time, one could mimik the stopped sigma-algebra construction, that is $\{A\in\mathcal{F}, \{N = i\}\cap A\in\sigma(X_1,\ldots,X_i)\}$, where $\mathcal{F}$ is the whole sigma-algebra of events. Here the obstacle is perhaps the non-measurability of $N$ with respect to the filtration though.