We observe the satisfaction of the sum rule for the conductivity of neutral fermions in an optical lattice subject to weak harmonic confinement. We measure the conductivity spectrum of the atoms through observations of the global current response to a perturbative applied force, using a quantum gas microscope. The spectrum is measured up to frequencies sufficient to characterize intraband transport, but well below the bandgap. The spectral weight of the response satisfies the sum rule in the limit of small lattice depth, but diminishes as the depth increases, reflecting an increase in the band-averaged effective mass. Measured under varying temperatures, densities, interaction strengths, and lattice depths, the spectral weight is shown to be obtainable from a thermodynamic description of the system. Furthermore, it is shown to be unaffected by varying the strength of interactions between the fermions, illustrating a fundamental prediction for conductivity spectra. The spectral weight characterizes the strength of the current response to an impulse, and therefore underpins the resistivity. As our measurements approach a high-temperature regime, its inverse is shown to approach T-linear behaviour.