When I was reading one mathematician's blog (I forgot his name by now), I encountered with the opinion that the measure theory should be studied without any $\sigma$-algebras and so on. Instead one should use the language of commutative Von Neuman algebras. I am not suggesting to discuss this opinion I rather want to ask if there are some textbooks treating measure theory in such a way.
2026-03-30 04:20:41.1774844441
Measure theory with algebraic point of view
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It's not quite the full treatment that you might want. But $\textit{Analysis Now}$ by Gert Pedersen has a chapter at the end where he develops measure theory by starting with positive linear functions on spaces of functions and getting Radon measures afterwards.