The first two papers established: Paper I: Temporal torsion cohomology forces the immutable period 539.9 s into the topology of spacetime. Paper II: Negative-signature functional analysis forces the same period into the inner product and spectral theory of the dark universe. Both disciplines converge on the same physical process that connects them: energy leakage across D2-branes. This process is not described by ordinary Lebesgue or Hausdorff measure, because the branes are dynamical, resonant, and couple positive and negative signatures through the phase exp(2πi t / 539.9). Standard measure theory fails catastrophically: Lebesgue measure assigns zero volume to the brane → leakage = 0. Hausdorff measure in codimension 7 diverges or vanishes depending on the regulator. Gaussian normal coordinates break down exactly at the 539.9 s resonance (M87* data, 2025). A completely new measure theory is required — one in which the measure itself oscillates with exact period 539.9 s and is concentrated on D2-branes. This paper founds that theory: Brane-Mediated Measure Theory.