Regarding the phase space density of stars and the Maxwell-Boltzmann distribution

Question

I was reading a paper where the authors effectively made the following equality when talking about stellar populations:

$$\frac{\mathrm{d} N_* }{ \mathrm{d} m_* \mathrm{~d}^3 \mathbf{x} \mathrm{d}^3 \mathbf{v}_*} = f\left(\mathbf{v}_*\right) n_{m_*}\tag{1}$$

where $\frac{\mathrm{d} N_* }{ \mathrm{d} m_* \mathrm{~d}^3 \mathbf{x} \mathrm{d}^3 \mathbf{v}_*}$ is the phase space density of stars - number of stars per unit spatial volume per unit velocity space volume per unit mass interval; $f\left(\mathbf{v}_*\right)$ is the Mawell-Boltzmann velocity distribution and $n_{m_*} = \frac{\mathrm{d} N_*}{\mathrm{d} m_* \mathrm{~d}^3 \mathbf{x}}$. If I rewrite the phase space density as $p(m_*, \mathbf{x}, \mathbf{v}_*)$, i wanted to know why the following (which is rewriting eqn (1)) holds true:

$$p(m_*, \mathbf{x}, \mathbf{v}_*) = f\left(\mathbf{v}_*\right) \int p(m_*, \mathbf{x}, \mathbf{v}_*)\mathrm{d}^3 \mathbf{v}_*$$

Answer 1

The assumption in equation (1) is that the velocity distribution is the same at every position and for every star mass. That is not true for a realistic system and can only have been a simplifying approximation.