# Cardinality of a $\sigma$-algebra over a set of independent random variables.

Problem statement

Let us say that we have some set of independent random variables: $$X = \{X_i\}^n_{i=1}$$ defined over a probability space of $$~(\Omega, \mathcal{F}, P)$$. I want to understand whether the following holds:

If $$\mathcal{F}_X = \sigma\left(\left\{X_1, X_2, \dots, X_n\right\}\right)$$, then is it, in general, true that: $$\left|\mathcal{F}_X\right| = \sum\limits_{\forall i \in I_n}|\sigma(\{X_i\})|$$?

As we know (and as proposed here), as $$X$$ consists of independent R.V.-s, then we can state that $$\forall i,j \in I_n:\sigma(\{X_i\})$$ and $$\sigma(\{X_j\})$$ are independent. But how to proceed from this fact to the split of the cardinality of $$\mathcal{F}_X$$?

I would appreciate any help, thank you in advance!

Related questions:

• The comment section to the linked question about the sigma algebra of two independent random variables already contains a counter example. If it doesn't hold for two, then there is no reason to believe that it would hold for general $n$. Nov 5 '21 at 23:45
• @LeanderTilstedKristensen oh, that's true, for some reason, I didn't notice it at the first look. It seems, that in the case of mine, the general case can be disproven by a simple counter-example! Thank you for your note! Nov 5 '21 at 23:50