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Spectrum sharing, as an approach to significantly improve spectrum efficiency in the era of 6th generation mobile networks (6G), has attracted extensive attention. Radio Environment Map (REM) based low-complexity spectrum sharing is widely studied where the spectrum occupancy measurement (SOM) is vital to construct REM. The SOM in three-dimensional (3D) space is becoming increasingly essential to support the spectrum sharing with space-air-ground integrated network being a great momentum of 6G. In this paper, we analyze the performance of 3D SOM to further study the tradeoff between accuracy and efficiency in 3D SOM. We discover that the error of 3D SOM is related with the area of the boundary surfaces of licensed networks, the number of discretized cubes, and the length of the edge of 3D space. Moreover, we design a fast and accurate 3D SOM algorithm that utilizes unmanned aerial vehicle (UAV) to measure the spectrum occupancy considering the path planning of UAV, which improves the measurement efficiency by requiring less measurement time and flight time of the UAV for satisfactory performance. The theoretical results obtained in this paper reveal the essential dependencies that describe the 3D SOM methodology, and the proposed algorithm is beneficial to improve the efficiency of 3D SOM. It is noted that the theoretical results and algorithm in this paper may provide a guideline for more areas such as spectrum monitoring, spectrum measurement, network measurement, planning, etc.