It seems something simple but I still haven’t totally grasped.
While reading this wikipedia’s derivation of of Chi-Squared distribution with k-degrees of freedom:
Consider the $k$ samples $x_i$ to represent a single point in a $k$-dimensional space. The chi square distribution for $k$-degrees of freedom will then be given by:
$$ P(Q) \, dQ = \int_\mathcal{V} \prod_{i=1}^k (N(x_i)\,dx_i) = \int_\mathcal{V} \frac{e^{-(x_1^2 + x_2^2 + \cdots +x_k^2)/2}}{(2\pi)^{k/2}}\,dx_1\,dx_2 \cdots dx_k $$
where $N(x)$ is the standard normal distribution and $\mathcal{V}$ is that elemental shell volume at $Q(x)$ which is proportional to the $(k-1)$-dimensional surface in $k$-space for which
$Q=\sum_{i=1}^k x_i^2$
The derivation continues, but I have a question concerning the term $dQ$ in the above equation for $P(Q)dQ$.
What I understand of the probability of $P(Q)$ is that it is a composition of the function $Q: \mathbb{R}^k \to \mathbb{R}$ with the probability function $P$. So the probability of the event $P(Q = q)$ is the probability of the event that is a surface of the $k$-sphere which is the level surface of $w = Q$ at $q$ (which is, the surface $Q(x) = q$).
So the probability of $P(Q = q)$ should be the integral over this spherical surface of the density function (this density function being $k$-independent joint normal distributions multiplied). However I don’t quite get de $dQ$ term. Shouldn’t it be just $P(Q)$ ?