Measurable set : some intuition in different definition. I'm a little bit in truble with definitions. Let $(X,\mathcal F,\mu)$. I have that a set $A\subset X$ is measurable if 
$1)$ $A\in \mathcal F$. 
$2)$ An other definition tels me that $A$ is measurable if Caratheodory formula hold, i.e. $$\mu(E)=\mu(E\cap A)+\mu(E\cap A^c)$$for all $E\subset X$. 
Let consider $(\mathbb R,\mathcal B,m)$ where $\mathcal B$ are the Borels set and $m$ the Lebesgue measure. Here I also have an other definition :
$3)$ For all $\varepsilon>0$ there is an open set $\mathcal O\supset A$ s.t. $$m(\mathcal O\backslash A)<\varepsilon.$$
I proved that 2) and 3) are equivalent. But the problem is we now that there are measurable set that are not Borel set. So typically, in $(\mathbb R,\mathcal B,m)$, definitions 2) and 3) doesn't hold, right ? So when do I have to use $2)$ and $3)$ and when do I have to use $1)$ ? (btw, I think that $1)$ always work)... So definitions $2)$ and $3)$ are wrong ?
 A: Definition (1) is the definition of "measurable" when you have an arbitrary, abstract measure space.  A measure space is just a triple $(X,\mathcal F,\mu)$, and when we have such a triple, we refer to the elements of $\mathcal{F}$ as the "measurable sets".
Definitions (2) and (3) are special cases of this: they are definitions of "measurable" in specific situations that provide a definition of the $\sigma$-algebra $\mathcal{F}$ for some particular measure space.  Definition (2) applies when you have a set $X$ with an outer measure $\mu$ on it.  Then if we define $\mathcal{F}$ as the collection of all sets that are "measurable" according to definition (2), the triple $(X,\mathcal{F},\mu|_{\mathcal{F}})$ is a measure space.
Definition (3) again provides a definition of a $\sigma$-algebra $\mathcal{F}$, consisting of all subsets $A$ of $\mathbb{R}$ satisfying this definition (such sets are typically called "Lebesgue measurable").  As you mention, this $\sigma$-algebra is not the same as the Borel $\sigma$-algebra $\mathcal{B}$, so this $\mathcal{F}$ consists of the measurable subsets of a different measure space: $(\mathbb{R},\mathcal{F},m')$ rather than $(\mathcal{R},\mathcal{B},m)$ (where $m'$ is the natural extension of $m$ to sets satisfying definition (3)).
The main point here is that the term "measurable" only has meaning in the context of a specific measure space, and its meaning depends on which measure space you are talking about.  Moreover, while definition (1) presents a measure space as given and then "measurable" defined in terms of that, in practice we often write things the other way around.  We first define a collection of sets to be "measurable", and then use that collection as the $\sigma$-algebra to define some measure space.
