= Solution
For $1\leq k\leq m$, the <correlation function of a point process> associated with the <eigenvalues> is
$$
R_k(x_1,\ldots,x_k)=\frac{m!}{(m-k)!}\int_{\mathbb R^{m-k}}f_m(x_1,\ldots,x_m)\,dx_{k+1}\cdots dx_m.
$$
The <factorial> factor counts ordered selections of $k$ distinct <eigenvalues>, so this is a <factorial> moment density, rather than the ordinary probability density of $k$ particular labels. Equivalently, for a nonnegative measurable test <function> $F$,
$$
\mathbb E\sum_{i_1,\ldots,i_k\text{ distinct}}F(\lambda_{i_1},\ldots,\lambda_{i_k})=\int_{\mathbb R^k}F(x_1,\ldots,x_k)R_k(x_1,\ldots,x_k)\,d\mathbf x.
$$
Repeated <projection kernel determinant integration>, with $r=m$, gives a factor $(m-k)!$ on integrating an $m$-by-$m$ kernel <determinant> down to size $k$. Together with the normalizing $1/m!$ in $f_m$, the factors cancel:
$$
\boxed{R_k(x_1,\ldots,x_k)=\det[K_m(x_i,x_j)]_{i,j=1}^k.}
$$
In particular $R_1(x)=K_m(x,x)$ and $\int R_k\,d\mathbf x=m!/(m-k)!$. Set $R_0=1$ and $R_k=0$ for $k>m$. The <eigenvalue> configuration is thus a <determinantal point process> with a rank-$m$ <finite-rank projection kernel>.
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