Integrals Preserve Order, Symmetry and Positive Semidefiniteness

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$. For $E\subseteq\mathbb{R}$, a map $g:E\to\mathbb{R}$ and $u,v\in E$, the oriented integral $\int_{u}^{v}g$ and what it means that it is defined are as in The Riemann Integral on a Closed Interval via Lower and Upper Integrals (vii) and The Riemann Integral on a Closed Interval via Lower and Upper Integrals (viii). For $a,b\in\mathbb{R}$ with $a<b$, a map $M:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ and $u,v\in[a,b]$, the integral $\int_{u}^{v}M=\int_{u}^{v}M(r)\,dr$ and what it means that it is defined are as in Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv), and $M$ is continuous on $[a,b]$ if it is continuous on $[a,b]$ with $d_{\mathbb{R}}$ on $[a,b]$ and $d_{n}$ on $\mathcal{M}_{n}(\mathbb{R})$. The set $\mathcal{S}(n)$ of symmetric matrices, positive semidefinite matrices and the ordering $\preceq$ are as in Euclidean Space and Real Matrices: Standing Notation (iv) and Euclidean Space and Real Matrices: Standing Notation (v).

(i) Let $E\subseteq\mathbb{R}$, let $f,g:E\to\mathbb{R}$, and let $u,v\in E$ with $u\le v$ be such that $\int_{u}^{v}f$ and $\int_{u}^{v}g$ are defined. If $f(r)\le g(r)$ for every $r\in E$ with $u\le r\le v$, then $\int_{u}^{v}f\le\int_{u}^{v}g$.

In the remaining clauses, let $a,b\in\mathbb{R}$ with $a<b$, let $u,v\in[a,b]$, and let $M,N:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be continuous on $[a,b]$ with $M(r)\in\mathcal{S}(n)$ and $N(r)\in\mathcal{S}(n)$ for every $r\in[a,b]$.

(ii) $\int_{u}^{v}M$ is defined and $\int_{u}^{v}M\in\mathcal{S}(n)$.

(iii) If $u\le v$ and $M(r)$ is positive semidefinite for every $r\in[a,b]$ with $u\le r\le v$, then $\int_{u}^{v}M$ is positive semidefinite.

(iv) If $u\le v$ and $M(r)\preceq N(r)$ for every $r\in[a,b]$ with $u\le r\le v$, then $\int_{u}^{v}M\preceq\int_{u}^{v}N$.