Null sets in $\mathbb{R}$ We know $A \subseteq \mathbb{R}$ is null if given $\epsilon &gt; 0$, there exists intervals     $\{I_n\}_{n \geq 1}$ such that 

$$ A \subseteq \bigcup_{n=1}^{\infty} I_n    \text{    and   }    \sum|I_n| &lt; \epsilon $$

My question is that if we change the words interval in the above definition with open intervals, closed intervals ,and intervals of the form $(a,b]$, $[a,b)$, then we get an equivalent notion?
If you are dealing with Borel $\sigma$-field over $\mathbb{R}$ then for any property $P$ that depends only on the measure defined on that space, if $P$ holds for any one of the following class of sets, then it holds for any other class, and also for the entire Borel $\sigma$-field.


class of all intervals,
class of all open intervals,
class of all closed intervals,
class of all clopen intervals,
class of all half-lines etc.
If you are dealing with Borel $\sigma$-field over $\mathbb{R}$ then for any property $P$ that depends only on the measure defined on that space, if $P$ holds for any one of the following class of sets, then it holds for any other class, and also for the entire Borel $\sigma$-field.


class of all intervals,
class of all open intervals,
class of all closed intervals,
class of all clopen intervals,
class of all half-lines etc.