I'm reading the first chapters of Landau & Lifshitz's Statistical Physics and I don't understand the definition of the quantum microcanonical ensemble. 

The microcanonical distribution for a quantum isolated system with energy $E\_0$ is defined in the text as:

$$\text d w = \text{const}\times \delta (E-E\_0) \cdot \prod \_a \text d \varGamma\_a.$$

where $\text d w$ stands for “probability” and $\text d \varGamma\_a\!$ is the differential of the function $\varGamma\_a(E\_a)$, the number of quantum states of a subsystem with with $\text {energies}\leq E\_a$ (which is treated as a smooth function).

I see two problems here: first of all, the distribution should be given in terms of a density operator $\rho$: I don't immediately see how to relate this density function to a matrix. However, forgetting for a second about the operator, what does $\text d w = ...$ mean? It is the probability density of *what variables* and how do these variables identify the state of the ensemble?

Finally, the text goes on asserting that, if the system is subdivided in $n$ small (quasi-)independent subsystems of the original system, then $\text d \Gamma$ can be written as the product $\text d \Gamma\_1\cdots \text d \Gamma \_n$. This fact is used later to derive the second law of thermodynamics: the joint probability density for the values $E\_1 , \dots ,E\_n$ of the subsystems energy is written as 

$$\qquad\qquad\qquad\qquad\qquad \text d w = \text{constant} \times \delta (E-E\_0)\cdot\prod\_a \dfrac{\text d \varGamma\_a}{\text d E\_a} \text d E\_a. \quad\qquad\qquad(7.15)$$

This one looks like the meaningful equation (to express the joint probability density of energies in the microcanonical ensemble). Is Landau's notation just a short-hand for the above one?