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

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:

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