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1: 1.4 Calculus of One Variable

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§1.4(i) Monotonicity

►If $f(x_1)\le f(x_2)$ for every pair $x_1$, $x_2$ in an interval $I$ such that , then $f(x)$ is nondecreasing on $I$. … ►If $f^{\prime }(x)\ge 0$ ($\le 0$) ($=0$) for all $x\in (a,b)$, then $f$ is nondecreasing (nonincreasing) (constant) on $(a,b)$. … ►A generalization of the Riemann integral is the Stieltjes integral ${\int }_a^{b}f(x)d\alpha (x)$, where $\alpha (x)$ is a nondecreasing function on the closure of $(a,b)$, which may be bounded, or unbounded, and $d\alpha (x)$ is the Stieltjes measure. … ►For $\alpha (x)$ nondecreasing on the closure $I$ of an interval $(a,b)$, the measure $d\alpha$ is absolutely continuous if $\alpha (x)$ is continuous and there exists a weight function $w(x)\ge 0$, Riemann (or Lebesgue) integrable on finite subintervals of $I$, such that …

2: 1.16 Distributions

… ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: …

3: 2.8 Differential Equations with a Parameter

… ►The regions of validity ${𝚫}_j({\alpha }_j)$ comprise those points $\xi$ that can be joined to ${\alpha }_j$ in $𝚫$ by a path ${𝒬}_j$ along which $\mathrm{\Re }v$ is nondecreasing $(j=1)$ or nonincreasing $(j=2)$ as $v$ passes from ${\alpha }_j$ to $\xi$. …

4: 18.2 General Orthogonal Polynomials

… ►More generally than (18.2.1)–(18.2.3), $w(x)dx$ may be replaced in (18.2.1) by $d\mu (x)$, where the measure $\mu$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to a bounded nondecreasing function $\alpha$ on the closure of $(a,b)$ with an infinite number of points of increase, and such that for all $n$. …