How to compare size of $\sigma$-algebras on $\mathbb{R}$, or more generally sets of sets of $\mathbb{R}$?

Question

I understand that this is a peculiar question that might not be very enlightening, but I am trying to compare how much bigger the set of Lebesgue measurable sets are compared to the set of Borel sets, and compare both of these sets to the power set of $\mathbb{R}$.

Using https://faculty.etsu.edu/gardnerr/5210/notes/Cardinality-of-M.pdf, I can see that $|\mathcal{L}| = |\mathcal{P}(\mathbb{R})|$. Using cardinalities, how can we compare the size of the Borel $\sigma$-algebra?

Another question I had in mind is if it is possible to construct a measure on sets of sets of $\mathbb{R}$. In other words, how can I construct an outer measure $\mu*: \mathcal{P}(\mathcal{P}(\mathbb{R})) \to [0, \infty]$? This way could be a different way to compare the size of sets, but I cannot come up with any natural and/or nice measures.

For the cardinality of the Borel $\sigma$-algebra: https://math.stackexchange.com/questions/70880/cardinality-of-borel-sigma-algebra — arkeet, Jul 02 '24 at 01:44
See Probability of selecting a non-measurable set -- the answers, comments, other cited questions in the answers/questions, and the Linked questions on the right side of "Probability of selecting a non-measurable set". Also relevant is Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras? — Dave L. Renfro, Jul 02 '24 at 02:01
@DaveL.Renfro Thank you! The topological argument is quite neat. — Sai Nallani, Jul 02 '24 at 03:40

How to compare size of $\sigma$-algebras on $\mathbb{R}$, or more generally sets of sets of $\mathbb{R}$?

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