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Is a countable union of measurable sets measurable? If the sum of measures of those measurable sets is finite, then their union is also measurable. But if the sum of measures of measurable sets is infinite, then the measure of their union may or may not be finite. Is their union still measurable?

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Yes, it is. The measurable sets form a $\sigma$-algebra, so are closed under countable unions (and intersections). Having a finite measure is not part of the definition of being measurable; it's the reason why we allow the value $+\infty$ for measures.

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