Explaining Klauder's "Modern Approach" Convergence Question

[ManDay]

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

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?

 

[mmwave]

The reformulation in the forum post is using some mathematical tools and theorems to simplify the original integral and make it easier to analyze. Let's break it down step by step.

First, the author uses the fact that the limit r → 0 exists and leads to an integral over all of ℝ. This means that as r approaches 0, the integral will cover all values on the real number line. This is a common assumption in mathematical analysis and allows us to simplify the integral by removing the limit notation.

Next, the author uses some properties of integrals to rewrite the original integral as a sum of two integrals. The first integral is over all values of x that are less than or equal to r, denoted by |x|≤r, and the second integral is over all values of x that are greater than r, denoted by |x|>r.

Then, the author uses the linearity property of integrals to pull out the factor of r^-1 from the first integral. This is a common technique in mathematical analysis and allows us to rewrite the integral in a simpler form.

Next, the author uses the fact that r^-1μr(x) is a probability measure to rewrite the first integral as an integral over the entire real number line, with the added term of itx/(1+x^2). This term is derived from the definition of the probability measure and its properties.

The author then uses some basic algebraic manipulations to further simplify the integral. This involves separating the term containing t and the term without t.

Finally, the author introduces a new measure σ(x) to represent the second integral over |x|>r. This measure satisfies the condition that the integral of x^2/(1+x^2) over all values of x greater than 0 is finite. This is important because it ensures that the integral will converge and the result will be well-defined.

Overall, the author is using some basic properties of integrals and measures to simplify the original integral and make it easier to analyze. The necessary convergence is provided by the properties of the measures and theorems of mathematical analysis.