# Expected value as a function of quantiles?

I was wondering where there is a general formula to relate the expected value of a continuous random variable as a function of the quantiles of the same r.v. The expected value of r.v. $X$ is defined as:
$E(X) = \int x dF_X(x)$ and quantiles are defined as : $Q^p_X = \{x : F_X(x) = p \} =F_X^{-1}(p)$ for $p\in(0,1)$.

Is there for instance a function function $G$ such that: $E(X) = \int_{p\in(0,1)} G(Q^p_X) dp$

The inverse (right inverse in discrete case) of the cumulative distribution function $$F(x)$$ is called the quantile function, often denoted $$Q(p)=F^{-1}(p)$$. The expectation $$\mu$$ can be given in terms of the quantile function (when the expectation exists ...) as $$\mu=\int_0^1 Q(p)\; dp$$ For the continuous case, this can be showed via a simple substitution in the integral: Write $$\mu = \int x f(x) \; dx$$ and then $$p=F(x)$$ via implicit differentiation leads to $$dp = f(x) \; dx$$: $$\mu = \int x \; dp = \int_0^1 Q(p) \; dp$$ We got $$x=Q(p)$$ from $$p=F(x)$$ by applying $$Q$$ on both sides.
For the general case, we can interpret $$E(X) = \int x dF_X(x)$$ as a Riemann–Stieltjes integral which, when the range of $$X$$ is the finite interval $$[a,b]$$, is defined as a limit of approximating sums of the form, for partitions $$a=x_0, $$\sum_{i=0}^{n-1} x_i^* \left[F(x_{i+1})-F(x_i)\right]$$ Passing to the quantile function $$F^*$$ (read the $$^*$$ as indicating a generalized inverse), and using $$p_i=F(x_i)$$, $$p_i^*=F(x_i^*)$$ this goes over to the approximating sum $$\sum_{i=0}^{n-1} F^*(p_i^*) \left[ p_{i+1} - p_i \right]$$ which are approximating sums for $$\int_0^1 F^*(p) \; dp$$ Then the infinite range case is treated the same way as with Riemann integrals.