A dielectric resonator with a gap between the top plate and the rest has been useful for measuring the penetration depth (λ) of superconductor films, a parameter essential for obtaining the intrinsic microwave surface resistance (Rs) of thin superconductor films. We investigated effects of a gap on the microwave properties of TE0ml-mode sapphire resonators with a gap between the top plate and the rest of the resonator. Regardless of a 10 µm-gap in TE0ml-mode sapphire resonators, variations of the TE0ml-mode resonant frequency on temperature (Δf0) as well as TE0ml-mode unloaded Q remained almost the same due to lack of axial currents inside the resonator and negligible radiation effects. The λ of YBa2Cu3O7-δ (YBCO) films obtained from a fit to the temperature-dependent Δf0 appeared to be 195 nm at 0 K and 19.3 GHz, which was well compared with the corresponding value of 193 nm at 10 kHz measured by the mutual inductance method. The intrinsic Rs of YBCO films on the order of 1 mΩ, and the tan δ of sapphire on the order of 10-8 at 15 K and 40 GHz could be measured simultaneously using sapphire resonators with a 10 µm-gap.