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I am currently learning about measure theory and stumble upon the following:

Given the measure space $(X, \mathbb{E}, \mu)$ $$\mathbb{A} = \{ A \subseteq X \mid \exists E \in \mathbb{E} \space \text{so}\space A\subseteq E \space \text{and} \space\mu(E)=0 \}$$

I have a lot of small questions, as I don't fully understand what the above says, and the more I look at it, the more confused I get. What does it mean when $\mathbb{A}$ is defined with curly brackets?

Is it correct to say that $A \in \mathbb{A}$? Since $A\subseteq X$, does that automatically means $X \in \mathbb{A}$? No, right? But why?

Since $A \subseteq E$, and $E \in \mathbb{E}$. Does this mean that $A \in \mathbb{E}$ ? My guess is 'yes', because $A \subseteq X$ and $X \subseteq \mathbb{E}$ as $\mathbb{E}$ must contain all kinds of subsets that can be made from X

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  • $\begingroup$ If $X \in \mathbb{A}$ this means $X$ has a supserset of measure $0$. Is this true? $\endgroup$ – Randall Dec 2 '19 at 1:33
  • $\begingroup$ Perhaps you’re just curiously perusing the subject but it is pretty strange to go into measure theory without being familiar with set theory notation. Richard Silverman’s translation of Fomin & Kolmogorov’s Real analysis textbook has a good set theory intro and the second half is devoted to measure theory, by the way. $\endgroup$ – Nap D. Lover Dec 2 '19 at 3:32
  • $\begingroup$ Yes, I am aware that I am lacking necessary knowledge, before I can dive into measure theory, but this is for a mandatory (undergrad.) course I am currently taking. Thank you for your recommendation $\endgroup$ – mahma Dec 2 '19 at 16:54
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$A$ is not a particular set. $A$ and $E$ are “free variables” in the definition, and the definition would be the same if another letter (say $B$) were used in place of $A$. The definition of $\mathbb A$ only uses the symbols $A$ and $E$ to make it easier to explain what $\mathbb A$ is. The collection $\mathbb A$ is the collection of subsets of $X$ for which a certain thing is true about “the subset” — that thing being that there exists “a thing” in $\mathbb E$ that is a superset of “the subset” and where “the thing” has measure zero.

The curly bracket notation for defining sets is widely used in mathematics and should be familiar well before you begin to study measure theory. For example, the odd integers can be defined as the set $\{n\in\mathbb Z \,|\, \exists k\in\mathbb Z \mbox{ for which } n=2k+1\}$.

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