What is the space $L^p(\mathbb R)/_\sim$ where $f\sim g$ $\iff$ $f$ and $g$ has the same distribution? Let $L^1(\mathbb R)$ the set of function that are Lebesgue integrable. Define for $f$ and $g$ the relation $$f\sim g\iff m\{f\leq x\}=m\{g\leq x\},$$
where $m$ is the Lebesgue measure. It's an equivalence relation. How this equivalence relation is interesting ? In the probability point of view, it looks to be the space of random variable having the same law. Is this space important ? Commonly used ? Does someone knows reference for such a space ?

For example, a problem I see is the fact that $X:\Omega \to \mathbb R$ and $Y: \Omega '\to \mathbb R$ can be random variable on $(\Omega ,\mathcal F,\mathbb P)$ and $(\Omega ',\mathcal F',\mathbb P')$ (different probability space), but saying that $X$ and $Y$ are equivalent looks strange. So maybe, even if they are on the same probability space, at the end, $X\sim Y$ is not really relevant and doesn't give us interesting information. What do you think ?
 A: Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. For me, a random variable $X:\Omega \to \mathbb R$ with distribution $F$; that mean $$F(x)=\mathbb P(X\leq x)$$ is a representative class of the equivalent relation you gave. 
The intuition behind random variable is the that there are a lot of function such that $Y:\Omega \to \mathbb R$ is measurable and $$\mathbb P(Y\leq x)=F(x),$$
and there is no reason to choose one rather an other one to describe the problem. So, in some sense, taking a random variable $X:\Omega \to \mathbb R$ s.t. $\mathbb P(X\leq x)=F(x)$ is really : take one function "randomly" in $$\{X:\Omega \to \mathbb R\mid X\text{measurable and }F(x)=\mathbb P\{X\leq x\}\},$$
to describe the problem, and all will describes it correctly. Which exactly mean : take a representative of $[X]_F\in \mathbb F/_\sim$ where $$\mathbb F=\{X: \Omega\to \mathbb R \mid X\text{ measurable}\}$$
and $$X\sim Y\iff \mathbb P(X\leq x)=\mathbb P(Y\leq x),$$
and $[X]_F=\{X\in \mathbb F\mid \mathbb P(X\leq x)=F(x)\}.$
I think, this gives more intuition on why we call really them "Random variable" and not only measurable function (even if both are the same). As you can see, if $Z:\Omega \to \mathbb R$ is measurable, then $Z$ obviously follow the law $\mathbb P\{Z\leq x\}=:G(x)$, i.e. $Z\in [Z]_G$). But I think that to see a Random variable with distribution $F$ as a representative class of $[X]_F$ gives really the intuition of the fact that you don't have to know precisely what is $Z$ to describes the problem. 
Notice that in general, if you have a probability space $(\Omega ,\mathcal F,\mathbb P)$ and a random variable $X:\Omega \to \mathbb R$ you (surprisingly) can't necessarily gives $X$ under a nice form. Take for example $\Omega =(0,1)$, $\mathcal F$ the Borel sets of $(0,1)$ and $\mathbb P$ the Lebesgue measure on $(0,1)$. Then $$X(\omega ):=\inf\{x\in\mathbb R\mid F(x)\geq \omega \},$$
where $\displaystyle F(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty }^x e^{-\frac{x^2}{2}}\,\mathrm d x$ follow a normal distribution $\mathcal N(0,1)$, but you can't write $X(\omega )$ under a form such that it's easy to work with. But at the end, as far as such a random variable exist, knowing a closed form of such a random variable is irrelevant to describe the experience. 
