From Klauder's "Modern Approach to Functional Integration"... The integrals go over all of ℝ.(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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# Convergence question

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