Reference Request: Semi-Rings and Rings (System of Sets, not Algebraic Structures)

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I studied Probability Theory (from a Measure Theory viewpoint) using only Sigma-Algebras.

Recently, I got a book about measure theory that starts from Semi-Rings, but it's presentation is too compact. Anyone got a suggestion on a good book that shows measures on Semi-Rings?

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Chapter 3 of Aliprantis and Burkinshaw Principles of Real Analysis treats measures on semirings.