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

ISBN-10: 1483167305

ISBN-13: 9781483167305

ISBN-10: 1483194787

ISBN-13: 9781483194783

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9. THEOREM. } be a finite or enumerable aggre gate of Lebesgue sets no two of which have common points, and assume that their sum has a finite measure. , and f f(x)dx= f f(x)dz+ JLl+Lt+... JLX I f(z)dz+.... JLt Proof. ) 8λ(χ) λ if XGLX, xeRn-Lx. a; e L x + L2 + . . 8. 10. So far, only bounded functions and sets of finite measures have been admitted. Both these restrictions will now be removed. D E F I N I T I O N S . Let I b e a Lebesgue set of finite measure and f(x) a function which is Lebesgue-measurable in L, but not necessarily bounded.

7. THEOREMS. Let pv and qv denote the vectors of mo mentum and position of the vth particle of a system of masspoints, Fv be the force acting on it, mv its mass. ,n, where the functions /^(r, t) are continuous for r > 0, f** =fvX, / w = 0 and /^(O, t) = 0. Then the components of the 3-dimenn siónál vectors Σρν (the "total momentum" of the system), n n Σ (mv qv—pv t) and Σ

N . The proof follows from the theorem by induction. § 7c The Lebesgue measure I n statistical mechanics, often the ' Volumes'' or ''measures" of point sets have to be considered. The usual concept of measure attributed to Riemann or Peano is, however, not suitable: not every "reasonable'' point set need possess a Riemann measure and, in particular, the topological image of a set which is Riemann-measurable need not be Riemannmeasurable itself. But it is just this property (that the topo logical image of a measurable set be measurable) t h a t is an indispensable requirement for statistical mechanics.