Lebesgue integration

Proof. [Sketch] We leave it to the reader to check the first two criteria of Definition 3.2. As for Criterion 3, positive definiteness follows directly from the definition of the integral, while nondegeneracy can be deduced from the theory of Lebesgue integration, using the first equivalence relation defined in Section 3.1. The interested reader can work out the details in Exercise 3.9 or consult Rudin [Ru74, Theorem 1.39]. ... 

Note that these two functions are not equal in the usual sense. Using either Riemann or Lebesgue integration, show that for any function i/r —> C... 

It should be observed that, in physics, the elements xk of the composite real variable. Y = (xj, x2,..., x often themselves contain several variables. For instance, in the so-called coordinate picture, each element xk has the form xk = (rk, Ck), where rk is the three-dimensional coordinate for particle k and Ck is the coordinate for its spin, isotopic spin, etc., taking only discrete values. It should be observed that, in the relations given by Eqs. (2.3) and (2.4), the integration symbol J dX indicates Lebesgue integration over all space coordinates involved between — oo and +oo and summation over all the discrete variables. 

The latent-heat terms (3.112) become necessary whenever the integrand ACP undergoes discontinuous change at a phase transition, with accompanying release of hidden AH. [The latent heat contribution is automatically included if one understands J(ACP) dT as Lebesgue integration.] For numerical evaluation of the integral in (3.111), power series... 

Let dN E) = n E) dE be a given distribution in the interval [ , ] with n(E) a nonnegative function measurable in the Lebesgue s sense moreover, let N E) be absolutely continuous and well behaved at oo. The Stieltjes-Lebesgue integral... 

In the continuous case, we suppose the sample space is the Euclidean space R ", and assume there is a (normalized) probability measure d/z defined by the density p which is a non-negative function. In this case the standard event space F is then typically taken to be the Borel a-algebra of subsets of R" which includes open balls and countable unions, countable intersections or relative complements of open balls in R . The measure of the set can be defined by Lebesgue integration... 

In many cases we may assume the existence of a density 7t xo, z, t) such that the transition probability is given by Lebesgue integration ... 

Even if the reader is perhaps not a true-born mathematician, he need not be afraid of the name Lebesgue integral . It is just the integral he knows from practice, only precised mathematically so as to include a broad class of regions HI and functions / (for example not only continuous functions). At least intuitively, he then perhaps accepts also the following results. 

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