# Measure space and nullsets

#### Meelas

Hey,

I have the measure space $$\displaystyle (\mathcal{X},\mathbb{E},\mu)$$.

I am considering the paving $$\displaystyle \mathbb{N}_{\mu}$$ of all nullsets given by: $$\displaystyle \mathbb{N}_{\mu}= \left\{ N \in \mathcal{X} : \exists N \subseteq E \hspace{0,1cm} \text{and} \hspace{0,1cm} \mu(E)=0 \right\}$$

I am also considering $\mathbb{E}_{\mu}$ which is an extension of $$\displaystyle \mathbb{E}$$ by the nullsets: $$\displaystyle \mathbb{E}_{\mu}= \left\{ E \cup N : E \in \mathbb{E} \hspace{0,1cm} \text{and} \hspace{0,1cm} N \in \mathbb{N}_{\mu} \right\}$$

A paving is understood to be an arbitrary collection of subsets.

I wish to show that a) $$\displaystyle \mathbb{E} \subseteq \mathbb{N}_{\mu}$$ and b) $$\displaystyle \mathbb{N}_{\mu} \subseteq \mathbb{E}_{\mu}$$.

Here is my attempt:

a)

If $$\displaystyle E \in \mathbb{E}$$ then $$\displaystyle E \in \mathbb{E}_{\mu}$$ since $$\displaystyle \mathbb{E}_{\mu}$$ is an extension of $$\displaystyle \mathbb{E}$$.

Or I could say:

Assume that $$\displaystyle E \in \mathbb{E}$$. Then $$\displaystyle E \in \mathbb{E}_{\mu}$$ since $$\displaystyle \mathbb{E}_{\mu}$$ consists of elements $$\displaystyle E \cup N$$ where $$\displaystyle E \in \mathbb{E}$$.

b)

Assume that $$\displaystyle N \in \mathbb{N}_{\mu}$$. Then $$\displaystyle N \in \mathbb{E}_{\mu}$$ since $$\displaystyle \mathbb{E}_{\mu}$$ consists of elements $$\displaystyle E \cup N$$ where $$\displaystyle N \in \mathbb{N}_{\mu}$$.

Both explanations a) and b) seem very simple and trivial (the "then" does not satisfy me)! But is the reasoning correct?

Appreciate corrections and guidance.