Does the measure of an increasing sequence ALWAYS equal to the limit of the measure of its subsequences?

Question

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 {A_n⊂A_n+1 for all n, then λ(lim_n→∞A_n) = λ(∪^∞_n=1A_n) = lim_n→∞(A_n).

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 A_n⊂B (class of measurable sets). My understanding is that this property is only true when A_n is a subset within a measurable space but not necessarily otherwise.

Answer 1

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