Integral of Positive Simple Function Well-Defined Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f: X \to \R, f \in \mathcal{E}^+$ be a positive simple function. Then the $\mu$-integral of $f$, $I_\mu \left({f}\right)$, is well-defined. That is, for any two standard representations for $f$, say: :$\displaystyle f = \sum_{i \mathop = 0}^n a_i \chi_{E_i} = \sum_{j \mathop = 0}^m b_j \chi_{F_j}$ it holds that: :$\displaystyle \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right) = \sum_{j \mathop = 0}^m b_j \mu \left({F_j}\right)$


Subrings of Integers are Sets of Integer Multiples/Examples/Even Integers Let $2 \Z$ be the set of even integers. Then $\struct {2 \Z, +, \times}$ is a subring of $\struct {\Z, +, \times}$.


Linear Second Order ODE/y'' + 8 y = 0 The second order ODE: :$(1): \quad y'' + 8 y = 0$ has the general solution: :$y = C_1 \cos 2 \sqrt 2 x + C_2 \sin 2 \sqrt 2 x$


Primitive of Reciprocal of x by Half Integer Power of a x squared plus b x plus c Let $a \in \R_{\ne 0}$. Then: :$\displaystyle \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n + \frac 1 2} } = \frac 1 {\paren {2 n - 1} c \paren {a x^2 + b x + c}^{n - \frac 1 2} } + \frac 1 c \int \frac {\d x} {x \paren {a x^2 + b x + c}^{n - \frac 1 2} } - \frac b {2 c} \int \frac {\d x} {\paren {a x^2 + b x + c}^{n + \frac 1 2} }$


Three Times Sum of Cubes of Three Indeterminates Plus 6 Times their Product :$3 \paren {a^3 + b^3 + c^3 + 6 a b c} = \paren {a + b + c}^3 + \paren {a + b \omega + c \omega^2}^3 + \paren {a + b \omega^2 + c \omega}^3$ where: : $\omega = -\dfrac 1 2 + \dfrac {\sqrt 3} 2$


Morphism in Preorder Category is Epic Let $\mathbf P$ be a preorder category. Let $f \in \mathbf P_1$ be a morphism. Then $f$ is epic.


Topologies on Set with More than One Element may not be Homeomorphic Let $S$ be a set which contains at least $2$ elements. Let $\tau_1$ and $\tau_2$ be topologies on $S$. Then it is not necessarily the case that $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic.


Either-Or Topology is Scattered Let $T = \struct {S, \tau}$ be the either-or space. Then $T$ is a scattered space.


Modified Fort Space is not Zero Dimensional Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not zero dimensional.


Equivalence of Definitions of Topologically Equivalent Metrics {{TFAE|def = Topologically Equivalent Metrics}} Let $M_1 = \struct {A, d_1}$ and $M_2 = \struct {A, d_2}$ be metric spaces on the same underlying set $A$.