On wikipedia, the following property of the Lebesgue measure is listed:

> 
> If $A$ is a disjoint union of countably many disjoint Lebesgue-measurable sets, then $A$ is itself Lebesgue-measurable and $\mu(A)$ is equal to the sum (or infinite series) of the measures of the involved measurable sets
> 
> 
> 

which I have interpreted as saying that

> 
> If $\lbrace \mathcal{X}\_i:i\in I\rbrace $ is a family of countably many sets, then let $A$ satisfy
>  $$
> A=\bigsqcup\_{i\in I}\mathcal{X}\_i
> $$
>  If $\mathcal{X}\_i$ and $\mathcal{X}\_j$ are disjoint for any $i,j\in I$, then $A$ is itself Lebesgue-measurable and $\mu(A)$ is equal to the sum (or infinite series) of the measures of the involved measurable sets
> 
> 
> 

Now, I am not certain if this interpretation is correct or not, so any corrections are appreciated, but above all else I find the fact that $A$ has to be the disjoint union rather redundant, since, if every $\mathcal{X}\_i$ is disjoint from $\mathcal{X}\_j$ for $i\neq j$, then $\mu(\mathcal{X}\_i\cup\mathcal{X}\_j)=\mu(\mathcal{X}\_i\sqcup\mathcal{X}\_j)$, so why bother with a disjoint union instead of a standard union? Is there a flaw in my logic?