In a system with a finite number of degrees of freedom $\eta_i$, $i=1,\ldots, N$ , the partition function depends on the N external fields that may couple linearly to the $\eta_i$ in the Hamiltonian $$ Z[H_i]= Tr \exp \left[ -\beta(\mathscr H - \sum_i H_i \eta_i \right] $$
In a system with infinite number of degrees of freedom, the partition function becomes a functional of $H(\mathbf r)$: $$ Z[H(\mathbf r)]= Tr \exp \left[ -\beta(\mathscr H - \int d^d \mathbf r H(\mathbf r) \eta(\mathbf r) \right] $$
What is meant by infinite number of degrees of freedom? Initially, there was a finite number (N) of axes in the system, and $\eta_i$ was constant throughout a given direction. I understand that for the nonhomogeneous case, $\eta$ has to depend on $\mathbf r$. But to do this, the author says that is due to an infinite number of degrees of freedom. In the continuum limit is the number of axes is infinity?