invariant measure - what does this notation mean?

Question

I am reading a paper and I am struggling to find what this notation means. I have given the link to the paper here (https://arxiv.org/abs/1108.0884).

$$\rho(d \mathbf{y}) = \frac{1}{\mathcal{Z}} \exp\left( -\sum_{k = N_0 + 1}^{N} \frac{\lambda_k}{\sigma^2 q_k^2} y_k^2\right) d \mathbf{y}$$

It is an invariant measure of multiple variables that are designated $y_k$ which are orthogonal. I don't know know what $d \mathbf{y}$ particularly with regard to integrating with respect to this measure. Any help that you could offer would be extremely useful.

Kindest regards,

Catherine

Answer 1

There are basically two notations commonly used for the "differential" in Lebesgue integration with respect to a variable $y$ and an explicitly defined measure $\rho$. One is $\rho(dy)$. The other is $d\rho(y)$. Neither is really very good, but generally everybody uses one or the other. Anyway, this differential is used in the sense that

$$\int_A \rho(dy) = \rho(A).$$

Thus in your example:

$$\rho(A)=\int_A \frac{1}{\mathcal{Z}} \exp\left( -\sum_{k = N_0 + 1}^{N} \frac{\lambda_k}{\sigma^2 q_k^2} y_k^2\right) d y$$