If space is empty, and if temperature measures how fast particles move, then how can space have a temperature?
PAUL LAFRENIERE, VERNER, ONTARIO
Space is not empty. It contains very tenuous gases, plasmas, and various types of radiation, like light, infrared, ultraviolet, X rays, etc. As you note, we're measuring the average speed of randomly moving particles when we measure temperature.
The solar wind, which consists of electrically charged gases (ions and electrons) flowing away from the Sun at 850,000 mph (1.3 million kilometers per hour) on average, permeates interplanetary space. Near Earth, each cubic centimeter of space a volume equivalent to that of a sugar cube contains between 100 and 10,000 of these particles.
In interstellar space, between the stars of our Milky Way Galaxy, we'd find about one particle in every cubic centimeter. In intergalactic space, between the galaxies of our universe, we would need to expand our sampling volume by a million times to a cubic meter, or 35 cubic feet in order to catch anywhere between 1 and 100 ions, electrons, and neutral atoms.