Doubt in Carathéodory's extension theorem

Question

I am new to the field of measure theory so I apologize before hand if this question is too trivial. It is stated that if $\Sigma_{0}$ is an algebra on set $S$ and $\mu_{0}$ is a positive measure/finitely additive map on $\Sigma_{0}$ . Let $\Sigma=\sigma(\Sigma_0)$, then there is a measure $\mu$ on $(S,\Sigma)$ such that $\mu(F)=\mu_0(F)$, $\forall$ F $\in \Sigma_0$ . I had a doubt like here $\Sigma$ is a $\sigma-$algbra on $S$ and that means it is closed under countable union and set complementation. So there must be infinite collection of subsets that belongs to $\Sigma$. But algebra $\Sigma_0$ is closed under finite intersection . The intuition is not clear since how can one extend a measure on finite collection (i.e. algebra) of subsets to a measure of infinite collection of subsets (i.e. in $\sigma-$algebra) ? It is very much intuitive that a property working on a subset of a set may not be true for the entire set. So how do one provides explanation for this ?