# Axiomatics of Classical Statistical Mechanics: International by Rudolf Kurth, I. N. Sneddon, S. Ulam, M. Stark

By Rudolf Kurth, I. N. Sneddon, S. Ulam, M. Stark

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4. THEOREM. / / the (bounded) function f(x) is Lebesguemeasurable in the Lebesgue set L of finite measure, then both f(x) and \f(x) \ are Lebesgue-integrable over each Lebesgue subset L' of L, and I f f(x)dxU f |/(s)|d*. IJL' I JL' Conversely, if the bounded function f(x) is integrable over the Lebesgue set L of finite measure, f(x) is Lebesgue-measurable in L. Proof. Cover the set of all values of f(x) by a sequence of half-open intervals of length €. To each interval, there corre­ sponds a Lebesgue subset of L such t h a t the values which f(x) takes in this set belong to the interval.

For example, the statistical inter­ pretation of the fundamental equations of hydromechanics in § 17. Outline of the proofs. 5. The statement con­ cerning the potential V(q, t) is proved by differentiating V(q, t) with respect to any coordinate qK: the result is the negative of the corresponding component of the force. 5 the last statement follows. § 11. Phase flow; Liouville's Theorem According to § 1, the first major difficulty presented by mechanical systems which have a large number of equations of motion is that, in general, the position of its initial point x in its phase space Γ cannot be actually observed.

If L is a Lebesgue set of finite measure and the functions f(x) and g(x) are Lebesgue-integrable over L, then also their sum f(x) + g(x) is integrable over L, and i {f(%) + 9(x)}dx = I f(x)dx+ JL JL JL g(x)dx. If both the Lebesgue sets Lx and L2 of finite measures have no common points, and if the function f(x) is integrable over Lx and L2, it is integrable over Σ± + Σ2) too, and f f(x)dx = ( f(x)dx+( J Lx+Lt JLX f(x)dx. f(x)dx = c. JL f(x)dx. /xL^ /(χ)άχ^ΜμΣ. THE LEBESGUE INTEGRAL 31 I n particular, the integral over a set of measure 0 always vanishes.