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ManDay

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From Klauder's "Modern Approach to Functional Integration"... The integrals go over all of ℝ.

What confuses me most is that this is pretty much at the beginning of the book and the author ocassionally explains rather obvious things but every now and then: something like this. Can anyone explain this reformulation and what theorems provide the necessary convergence?

Let µ_{r}be an r-dependent probability measure, r ∈ ℝ^{+}. By assumption, the limit r → 0 exists and leads to

$$\begin{align*}

\lim_{r\to0} \int ( 1 - e^{itx} ) r^{-1} \mathrm{d}\mu_r(x) & = \lim_{r\to0}\int \left( 1 - e^{itx} + \frac{itx}{1+x^2} \right) \mathrm{d}(r^{-1} \mu_r(x) ) - i t \lim_{r\to0} \int \frac x { 1 + x^2 } \mathrm{d} ( r^{-1} \mu_r(x) ) \\

& = - i a t + \lim_{r\to0} \int_{|x|\leq r } \left( 1 - e^{itx} + \frac {itx} {1+x^2} \right) \mathrm{d}(r^{-1} \mu_r(x) ) + \lim_{r\to0} \int_{|x|\gt r } \left( 1 - e^{itx} + \frac{ itx}{(1+x^2}\right) \mathrm{d}(r^{-1} \mu_r(x) ) \\

& = - i a t + b t^2 + \lim_{r\to0} \int_{|x|\gt r } \left( 1 - e^{itx} + \frac { itx }{ 1+x^2 }\right) \mathrm{d}\sigma ( x )

\end{align*} $$

where a ∈ ℝ, b > 0 and σ(x) is a nonnegative measure that satisfies

$$\int_{|x|\gt0} \frac {x^2}{ 1 + x^2 } \mathrm{d}\sigma(x) \lt \infty$$

What confuses me most is that this is pretty much at the beginning of the book and the author ocassionally explains rather obvious things but every now and then: something like this. Can anyone explain this reformulation and what theorems provide the necessary convergence?

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