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I am just starting out on measure theory and will be grateful if someone could clarify this doubt of mine -

One of the properties of a measure λ(.) is given as if {An⊂An+1 for all n, then λ(limn→∞An) = λ(∪n=1An) = limn→∞(An).

Now, I understand that this property ensures that different measures within the measure space will lead to consistent limits. However, my question is - is this ALWAYS true, irrespective of whether An⊂B (class of measurable sets). My understanding is that this property is only true when An is a subset within a measurable space but not necessarily otherwise.

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$\cup_{n=1}^{\infty}A_n$ is measurable, so all $A_n$ are subsets of a measurable space, therefore there is no otherwise.

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