Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging

Let $n\ge1$ be a natural number and let $w=(w_1,\dots,w_n)$ be a point of the Euclidean space $\mathbb{R}^{n}$, with $\lVert w\rVert_1=\sum_{j=1}^{n}|w_j|$, where $|\cdot|$ is the absolute value and sums over $1\le j\le n$ are the finite sums of the real numbers. Points of $[0,\infty)^{1+n}\subseteq\mathbb{R}^{1+n}$ are written $(a,b)=(a,b_1,\dots,b_n)$. Define

$$\mathsf{N}_w(a,b)=\sum_{j=1}^{n}w_j\,(a-b_j),\qquad \mathsf{D}(a,b)=\frac12\,a+\frac{1}{2n}\sum_{j=1}^{n}b_j ,$$

and the symmetrised score functional $\Phi_w:[0,\infty)^{1+n}\to[0,\infty)$ by

$$\Phi_w(a,b)=\frac{\mathsf{N}_w(a,b)^{2}}{\mathsf{D}(a,b)}\quad\text{if }\mathsf{D}(a,b)>0,\qquad \Phi_w(a,b)=0\quad\text{if }\mathsf{D}(a,b)=0 .$$

Measurability of real-valued maps on a measurable space is that of Measurable Function and Real-Valued Measurable Function with respect to the Borel $\sigma$-algebra of the real line; integrals of $[0,\infty]$-valued measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, and integrable means integrable. Sequential continuity on a subset of a Euclidean space is understood as in Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable, with respect to the Euclidean distance.

1. (Weighted Cauchy-Schwarz inequality) Let $(\mathsf{Z},\mathcal{Z},\nu)$ be a measure space, let $\beta:\mathsf{Z}\to[0,\infty)$ be measurable with $\int_{\mathsf{Z}}\beta\,d\nu<\infty$, and let $\alpha:\mathsf{Z}\to\mathbb{R}$ be measurable with $\alpha(z)=0$ for every $z$ with $\beta(z)=0$. Let $q:\mathsf{Z}\to[0,\infty)$ be the function equal to $\alpha^{2}/\beta$ on $\{\beta>0\}$ and to $0$ on $\{\beta=0\}$; then $q$ is measurable. If moreover $\int_{\mathsf{Z}}q\,d\nu<\infty$, then $\alpha$ is integrable and

$$\Bigl(\int_{\mathsf{Z}}\alpha\,d\nu\Bigr)^{2}\le\Bigl(\int_{\mathsf{Z}}\beta\,d\nu\Bigr)\Bigl(\int_{\mathsf{Z}}q\,d\nu\Bigr).$$

2. (Properties of the symmetrised score functional) (a) $\Phi_w$ is sequentially continuous on $[0,\infty)^{1+n}$. (b) For every $(a,b)\in[0,\infty)^{1+n}$,

$$0\le\Phi_w(a,b)\le2n\,\lVert w\rVert_1^{2}\Bigl(a+\sum_{j=1}^{n}b_j\Bigr).$$

(c) $\Phi_w(ca,cb)=c\,\Phi_w(a,b)$ for every real $c\ge0$, where $cb=(cb_1,\dots,cb_n)$. (d) If $a>0$ and $\delta$ is a real number with $0\le\delta\le1$ such that $b_j\ge(1-\delta)a$ for every $j$, then

$$\Phi_w(a,b)\le(1+\delta)\,\frac{\mathsf{N}_w(a,b)^{2}}{a}.$$

3. (Averaging inequality) Let $(\mathsf{Z},\mathcal{Z},\nu)$ be a measure space and let $\ell_0,\ell_1,\dots,\ell_n:\mathsf{Z}\to[0,\infty)$ be measurable with finite integrals $A=\int_{\mathsf{Z}}\ell_0\,d\nu$ and $B_j=\int_{\mathsf{Z}}\ell_j\,d\nu$ ($1\le j\le n$). Then the map $z\mapsto\Phi_w\bigl(\ell_0(z),\ell_1(z),\dots,\ell_n(z)\bigr)$ is measurable, and

$$\Phi_w(A,B_1,\dots,B_n)\le\int_{\mathsf{Z}}\Phi_w\bigl(\ell_0(z),\ell_1(z),\dots,\ell_n(z)\bigr)\,\nu(dz)\qquad\text{in }[0,\infty].$$