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

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.

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