Glasser's master theorem

In this posting we discuss a family of measure-preserving transforms on $\mathbb{R}$.

1. Glasser's master theorem

There is a complete characterization of rational functions which preserve the Lebesgue measure on $\mathbb{R}$. This appears at least as early as in Pólya and Szegö's book [3] in 1972, but seems more widely known as Glasser's master theorem since his 1983 paper [1]. Here we introduce a short proof of a sufficient condition.

Theorem. (Glasser, 1983) Let $a_1 < a_2 < \cdots < a_n$ and $\alpha$ be real numbers and $c_1, \cdots, c_n$ be positive real numbers. Then the function $$\phi(x) = x - \alpha - \sum_{k=1}^{n} \frac{c_k}{x - a_k}$$ preserves the Lebesgue measure on $\mathbb{R}$. In particular, for any Lebesgue-integrable function $f$ on $\mathbb{R}$ we have $$\int_{\mathbb{R}} f(\phi(x)) \, dx = \int_{\mathbb{R}} f(x) \, dx.$$

Proof. Let $I_k = (a_k, a_{k+1})$ for $k = 0, \cdots, n$ with the convention $a_0 = -\infty$ and $a_{n+1} = \infty$. Then by a direct computation, $\phi'(x) > 1$ on $\mathbb{R} \setminus \{a_1, \cdots, a_n\}$. Moreover, we have $$\phi(x) \to +\infty \quad \text{as} \quad x \to a_k^-, \qquad k = 1, \cdots, n+1$$ and similarly $$\phi(x) \to -\infty \quad \text{as} \quad x \to a_k^+, \qquad k = 0, \cdots, n.$$ This implies that $\phi$ is a bijection from $I_k$ to $\mathbb{R}$ for each $k = 0, \cdots, n$. Let $\psi_k : I_k \to \mathbb{R}$ be the inverse of the restriction $\phi|_{I_k}$ for each $k = 0, \cdots, n$, i.e., $\phi \circ \psi_k = \mathrm{id}_{\mathbb{R}}$. Then for each $y \in \mathbb{R}$, the equation $\phi(x) = y$ has exactly $n+1$ zeros $\psi_0(y), \cdots, \psi_n(y)$. Now multiplying both sides of this equation by $(x-a_1)\cdots(x-a_n)$, we obtain $$(x-\alpha-y)(x-a_1)\cdots(x-a_n) + \text{[polynomial in $x$ of degree $\leq n-1$]} = 0$$ Since the left-hand side is a polynomial of degree $n+1$, it follows that the left-hand side coincides with $(x-\psi_0(y))\cdots(x-\psi_n(y))$. Then comparing the coefficient of $n$-th term shows that $$y+\alpha+a_1+\cdots+a_n = \psi_0(y)+\cdots+\psi_n(y).$$ Therefore for each Lebesgue-measurable function $f$ on $\mathbb{R}$, we have $$\int_{\mathbb{R}} f(\phi(x)) \, dx = \sum_{k=0}^{n} \int_{I_k} f(\phi(x)) \, dx = \sum_{k=0}^{n} \int_{\mathbb{R}} f(y)\psi'_k(y) \, dy = \int_{\mathbb{R}} f(y) \, dy$$ and the proof is complete. ////

2. Examples

Example 1. The theorem above provides an easy way of computing the following integral: for $\alpha > 0$, $$\begin{align*} \int_{0}^{\infty} \exp\left\{ -x^2 - \frac{\alpha}{x^2} \right\} \, dx &= \frac{1}{2} \int_{-\infty}^{\infty} \exp\left\{ -x^2 - \frac{\alpha}{x^2} \right\} \, dx \\ &= \frac{1}{2} \int_{-\infty}^{\infty} \exp\left\{ -\left(x - \frac{\sqrt{\alpha}}{x} \right)^2 - 2\sqrt{\alpha} \right\} \, dx \\ &= \frac{1}{2} \int_{-\infty}^{\infty} \exp\{-x^2 - 2\sqrt{\alpha} \} \, dx \\ &= \frac{\sqrt{\pi}}{2} \exp\{-2\sqrt{\alpha} \}. \end{align*}$$

Example 2. (Lévy distribution) Let $c > 0$ and define $f : (0, \infty) \to \mathbb{R}$ by $$f(x) = \sqrt{\frac{c}{2\pi}} x^{-3/2} \exp\left\{-\frac{c}{2x}\right\}.$$ Its (probabilists') Fourier transform is defined as $$\phi(t) = \int_{0}^{\infty} e^{itx}f(x) \, dx.$$ Since $f$ is integrable, $\phi$ extends to a complex function which is holomorphic on the upper-half plane $\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0 \}$ and continuous on $\bar{\mathbb{H}}$. Now for $s > 0$, we have $$\begin{align*} \phi(is) &= \sqrt{\frac{c}{2\pi}} \int_{0}^{\infty} x^{-3/2} \exp\left\{-\frac{c}{2x}-sx\right\} \, dx \\ &= \frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \exp\left\{-u^2-\frac{cs}{2u^2}\right\} \, du, \qquad (x = (c/2)u^{-2}) \\ &= \exp\{-\sqrt{2cs}\}. \end{align*}$$ From this, we know that $$\phi(t) = \exp\{-\sqrt{-2ict}\}$$ initially along the imaginary axis with $\Im(t) > 0$. Since both sides extend to holomorphic functions on all of $\mathbb{H}$, this identity remains true on $\mathbb{H}$ by the principle of analytic continuation. Then by the continuity, this is also true for $t \in \mathbb{R}$.

3. Generalization

The following theorem generalizes the Glasser's master theorem. It tells that certain family of Nevanlinna functions gives rise to measure-preserving transformations on $\mathbb{R} = \partial \overline{\mathbb{H}}$.

Theorem. (Letac, 1977) Let $\alpha$ be a real number and $\mu$ be a measure on $\mathbb{R}$ which is singular to the Lebesgue measure and satisfies $\int_{\mathbb{R}} \frac{\mu(d\lambda)}{1+\lambda^2} < \infty$. Then the function $$\phi(x) = x - \alpha - \lim_{\epsilon \to 0^+} \int_{\mathbb{R}} \left( \frac{1}{x+i\epsilon - \lambda} + \frac{\lambda}{1+\lambda^2} \right) \, \mu(d\lambda)$$ defines a measruable function on $\mathbb{R}$ that preserves the Lebesgue measure on $\mathbb{R}$.

If $\mu$ is a finite sum of point masses, it reduces to the previous theorem.

References

• [1] Glasser, M. L. "A Remarkable Property of Definite Integrals." Mathematics of Computation 40, no. 162 (1983): 561-63. doi:10.2307/2007531.
• [2] Letac, Gérard. "Which Functions Preserve Cauchy Laws?" Proceedings of the American Mathematical Society 67, no. 2 (1977): 277-86. doi:10.2307/2041287.
• [3] G. Pólya and G. Szegö. "Problems and Theorems in Analysis" I, II, Problem 118.1 Springer-Verlag, Berlin and New York (1972).