Ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur–Tetrode equation for the entropy of an ideal gas and the Saha ionization equation for a weakly ionized plasma. The classical thermodynamic properties of an ideal gas can be described by two equations of state: The ideal gas law is the equation of state for an ideal gas, given by: where The ideal gas law is an extension of experimentally discovered gas laws. It can also be derived from microscopic considerations. Real fluids at low density and
Positive and negative sets union of a sequence of positive or negative sets is also positive or negative; more formally, if ("A") is a sequence of positive sets, then is also a positive set; the same is true if the word "positive" is replaced by "negative". A set which is both positive and negative is a μ-null set, for if "E" is a measurable subset of a positive and negative set "A", then both μ("E") ≥ 0 and μ("E") ≤ 0 must hold, and therefore, μ("E") = 0. The Hahn decomposition theorem states that for every measurable space ("X",Σ) with a signed measure μ,