hcju/proofwikips huggingface dataset:
Complex Plane is Complete Metric Space
The complex plane, along with the metric induced by the norm given by the complex modulus, forms a complete metric space.
Euclidean Space is Complete Metric Space
Let $m$ be a positive integer. Then the Euclidean space $\R^m$, along with the Euclidean metric, forms a complete metric space.
Complex Plane is Banach Space
The complex plane, along with the complex modulus, forms a Banach space over $\C$.
Rational Number Space is not Complete Metric Space
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean metric $d$. Then $\struct {\Q, \tau_d}$ is not a complete metric space.
Limit of Subsequence equals Limit of Sequence/Real Numbers
Let $\sequence {x_n}$ be a sequence in $\R$. Let $l \in \R$ such that: :$\displaystyle \lim_{n \mathop \to \infty} x_n = l$ Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\displaystyle \lim_{r \mathop \to \infty} x_{n_r} = l$
Distance Function of Metric Space is Continuous
Let $M = \struct {A, d}$ be a metric space. Let $\tau_A$ be the topology on $A$ induced by $d$. Let $\struct {A \times A, \tau}$ be the product space of $\struct {A, \tau_A}$ and itself. Then the distance function $d: A \times A \to \R$ is a continuous mapping.
Sequence Converges to Within Half Limit/Complex Numbers
Let $\sequence {z_n}$ be a sequence in $\C$. Let $\sequence {z_n}$ be convergent to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} z_n = l$ where $l \ne 0$. Then: : $\exists N: \forall n > N: \cmod {z_n} > \dfrac {\cmod l} 2$
Number to Reciprocal Power is Decreasing
The real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.
Inverse of Diagonal Matrix
Let: :$\mathbf D = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ \end{bmatrix}$ be an $n \times n$ diagonal matrix. Then its inverse is given by: : $\mathbf D^{-1} = \begin{bmatrix} \dfrac 1 {a_{11}} & 0 & \cdots & 0 \\ 0 & \dfrac 1 {a_{22}} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \dfrac 1 {a_{nn}} \\ \end{bmatrix}$ provided that none of the diagonal elements are zero. If any of the diagonal elements are zero, $\mathbf D$ is not invertible.  {{expand|If any diagonal element isn't a unit}}
Subspace of Complete Metric Space is Closed iff Complete
Let $\struct {M, d}$ be a complete metric space. Let $\struct {S, d}$ be a subspace of $\struct {M, d}$. Then $S$ is closed  $S$ is complete.
Set of Chains is Chain Complete for Inclusion
Let $\left({P, \leq}\right)$ be an ordered set. Let $\operatorname{ch} \left({P}\right)$ be the set of chains in $P$. Let $\mathcal C$ be a chain in $\operatorname{ch} \left({P}\right)$ with respect to the subset ordering. Then the union $\bigcup \mathcal C$ of $\mathcal C$ is also a chain in $P$.
Jensen's Inequality (Real Analysis)
Let $I$ be a real interval. Let $\phi: I \to \R$ be a convex function. Let $x_1, x_2, \ldots, x_n \in I$. Let $\lambda_1, \lambda_2, \ldots, \lambda_n \ge 0$ be real numbers, at least one of which is non-zero. Then: :$\displaystyle \map \phi {\frac {\sum_{k \mathop = 1}^n \lambda_k x_k} {\sum_{k \mathop = 1}^n \lambda_k} } \le \frac {\sum_{k \mathop = 1}^n \lambda_k \map \phi {x_k} } {\sum_{k \mathop = 1}^n \lambda_k}$ For $\phi$ strictly convex, equality holds  $x_1 = x_2 = \cdots = x_n$.
Real Linear Subspace Contains Zero Vector
Let $\mathbb W \subseteq \R^n$ such that $\mathbb W$ is a linear subspace of $\R^n$. Then $\mathbb W$ contains the zero vector: :$\mathbf 0_{n \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix} \in \mathbb W$
Zero Subspace is Subspace
Let $V$ be a vector space over $K$ with zero vector $\mathbf 0$. The zero subspace $\left\{{\mathbf 0}\right\}$ is a subspace of $V$.
Cartesian Product of Natural Numbers with Itself is Countable
The Cartesian product $\N \times \N$ of the set of natural numbers $\N$ with itself is countable.
Minkowski's Inequality for Sums
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers. Let $p \in \R$, $p \ne 0$ be a real number. If $p < 0$, then we require that $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be strictly positive. If $p > 1$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \le \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$ If $p < 1$, $p \ne 0$, then: :$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \ge \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$
Minkowski's Inequality for Integrals
Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$. {{explain|Riemann integrable or Lebesgue integrable?}} :$(1):\quad$ Let $p > 1$. Then: :: $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \le \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$ :$(2):\quad$ Let $p < 1, p \ne 0$. Then: :: $\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \ge \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$
Characteristic Function Determined by 1-Fiber
Let $A \subseteq S$. Let $f:S \to \left\{{0, 1}\right\}$ be a mapping. Denote by $\chi_A$ the characteristic function on $A$. Then the following are equivalent: :$(1):\quad f = \chi_A$ :$(2):\quad \forall s \in S: f \left({s}\right) = 1 \iff s \in A$ Using the notion of a fiber, $(2)$ may also be expressed as: :$(2'):\quad f^{-1} \left({1}\right) = A$
Characteristic Function of Intersection
Let $A, B \subseteq S$. Let $\chi_{A \cap B}$ be the characteristic function of their intersection $A \cap B$.
Characteristic Function of Union
Let $A, B \subseteq S$. Let $\chi_{A \mathop \cup B}$ be the characteristic function of their union $A \cup B$.
Characteristic Function of Set Difference
Let $A, B \subseteq S$. Then: :$\chi_{A \mathop \setminus B} = \chi_A - \chi_{A \cap B}$  where: :$A \setminus B$ denotes set difference :$\chi$ denotes characteristic function.
Characteristic Function of Symmetric Difference
Let $A, B \subseteq S$. Then: :$\chi_{A * B} = \chi_A + \chi_B - 2 \chi_{A \cap B}$  where: : $\chi$ denotes characteristic function : $*$ denotes symmetric difference.
Vector Subspace of Real Vector Space
Let $\R^n$ be a real vector space. Let $\mathbb W \subseteq \R^n$. Then $\mathbb W$ is a linear subspace of $\R^n$ : :$(1): \quad \mathbf 0 \in \mathbb W$, where $\mathbf 0$ is the zero vector with $n$ entries :$(2): \quad \mathbb W$ is closed under vector addition :$(3): \quad \mathbb W$ is closed under scalar multiplication.
Reverse Young's Inequality for Products
Let $p, q \in \R_{> 0}$ be strictly positive real numbers satisfying: :$\displaystyle \frac 1 p - \frac 1 q = 1$ Let $a \in \R_{\ge 0}$ be a positive real number. Let $b \in \R_{> 0}$ be a strictly positive real number. Then: :$\displaystyle a b \ge \frac {a^p} p - \frac {b^{-q}} q$
Reverse Hölder's Inequality for Sums
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that $\dfrac 1 p - \dfrac 1 q = 1$. Suppose that the sequences $\mathbf x = \sequence {x_n}$ and $\mathbf y = \sequence {y_n}$ in $\C$ (or $\R$) are such that the series :$\displaystyle \paren {\sum_{n \mathop = 1}^\infty \size {x_n}^p}^{1/p}$ and :$\displaystyle \paren {\sum_{n \mathop = 1}^\infty \size {y_n}^{-q} }^{-1/q}$ are convergent. Denote these values by $\norm {\mathbf x}_p$ and $\norm {\mathbf y}_{-q}$, respectively. Here, the notation $\norm {\mathbf x}_p$ does not denote a norm, but is instead just a convenient notation similar to that of the $p$-norm, which is only defined when $p \ge 1$. Let $\norm {\mathbf {x y} }_1$ denote the $1$-norm of $\mathbf {x y}$, if $\mathbf {x y}$ is in the Lebesgue space $\ell^1$. Then $\norm {\mathbf {x y} }_1 \ge \norm {\mathbf x}_p \norm {\mathbf y}_{-q}$.
Real Vector Space is Vector Space
Let $\R$ be the set of real numbers. Then the real vector space $\R^n$ is a vector space.
Sigma-Algebra Contains Empty Set
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Then $\varnothing \in \Sigma$.
Sigma-Algebra Closed under Union
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $A, B \in \Sigma$ be measurable sets. Then $A \cup B \in \Sigma$, where $\cup$ denotes set union.
Sigma-Algebra Closed under Countable Intersection
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Suppose that $\left({E_n}\right)_{n \in \N} \in \Sigma$ is a collection of measurable sets. Then $\displaystyle \bigcap_{n \mathop \in \N} E_n \in \Sigma$, where $\displaystyle \bigcap$ denotes set intersection.
Sigma-Algebra of Countable Sets
Let $X$ be a set. Let $\Sigma$ be the collection of countable and co-countable subsets of $X$. Then $\Sigma$ is a $\sigma$-algebra.
Trace Sigma-Algebra is Sigma-Algebra
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Let $E \subseteq X$ be a subset of $X$. Then the trace $\sigma$-algebra $\Sigma_E$ is a $\sigma$-algebra on $E$.
Pre-Image Sigma-Algebra on Domain is Sigma-Algebra
Let $X, X'$ be sets, and let $f: X \to X'$ be a mapping. Let $\Sigma'$ be a $\sigma$-algebra on $X'$. Then $f^{-1} \left({\Sigma'}\right)$, the pre-image $\sigma$-algebra on the domain of $f$, is a $\sigma$-algebra on $X$.
Intersection of Sigma-Algebras
Let $X$ be a set. Let $\family {\Sigma_i}_{i \mathop \in I}$ be a indexed set of $\sigma$-algebras on $X$. Then $\Sigma := \displaystyle \bigcap_{i \mathop \in I} \Sigma_i$ is also a $\sigma$-algebra on $X$. Here, $\displaystyle \bigcap$ denotes set intersection.
Generated Sigma-Algebra Preserves Subset
Let $X$ be a set. Let $\mathcal F, \mathcal G \subseteq \mathcal P \left({X}\right)$ be collections of subsets of $X$. Suppose that $\mathcal F \subseteq \mathcal G$. Then $\sigma \left({\mathcal F}\right) \subseteq \sigma \left({\mathcal G}\right)$, where $\sigma \left({\mathcal G}\right)$ denotes the $\sigma$-algebra generated by $\mathcal G$
Characterization of Euclidean Borel Sigma-Algebra
Let $\mathcal{O}^n$, $\mathcal{C}^n$ and $\mathcal{K}^n$ be the collections of open, closed and compact subsets of the Euclidean space $\left({\R^n, \tau}\right)$, respectively. Let $\mathcal{J}_{ho}^n$ be the collection of half-open rectangles in $\R^n$. Let $\mathcal{J}^n_{ho, \text{rat}}$ be the collection of half-open rectangles in $\R^n$ with rational endpoints. Then the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$ satisfies: :$\mathcal B \left({\R^n}\right) = \sigma \left({\mathcal{O}^n}\right) = \sigma \left({\mathcal{C}^n}\right) = \sigma \left({\mathcal{K}^n}\right) = \sigma \left({\mathcal{J}_{ho}^n}\right) = \sigma \left({\mathcal{J}^n_{ho, \text{rat}}}\right)$ where $\sigma$ denotes generated $\sigma$-algebra.
Generated Sigma-Algebra Preserves Finiteness
Let $X$ be a set, and let $A_1, \ldots, A_n \subseteq X$ be subsets of $X$. Then $\sigma \left({\left\{{A_1, \ldots, A_n}\right\}}\right)$ is a finite set, where $\sigma$ denotes generated $\sigma$-algebra.
Set of Invertible Mappings forms Symmetric Group
Let $S$ be a set. Let $\struct {S^S, \circ}$ be the monoid of all the mappings from $S$ to itself. Let $\mathcal G$ be the set of all invertible mappings from $S$ to $S$. Then $\struct {\mathcal G, \circ}$ is the symmetric group on $S$.
Group of Units of Field
Let $k$ be a field. Then $k^\times = k \setminus \set 0$
Automorphism Group is Subgroup of Symmetric Group
Let $\struct {S, *}$ be an algebraic structure. Let $\Aut S$ be the automorphism group of $\struct {S, *}$. Then $\Aut S$ is a subgroup of the symmetric group  $\struct {\Gamma \paren S, \circ}$ on $S$.
Differentiation of Vector-Valued Function Componentwise
Let: :$\mathbf r: t \mapsto \left\langle{r_1 \left({t}\right), r_2 \left({t}\right), \ldots, r_n \left({t}\right)}\right\rangle$  be a differentiable vector-valued function. The derivative of a vector-valued function can be calculated by differentiating each of its component functions: :$\dfrac {\d \mathbf r \left({t}\right)} {\d t} = \left\langle{D_t r_1 \left({t}\right), D_t r_2 \left({t}\right), \ldots, D_t r_n \left({t}\right)}\right\rangle$
Derivative of Dot Product of Vector-Valued Functions
Let: : $\mathbf r: x \mapsto \left\langle{r_1 \left({x}\right), r_2 \left({x}\right), \ldots, r_n \left({x}\right)}\right\rangle$ : $\mathbf q: x \mapsto \left\langle{q_1 \left({x}\right), q_2 \left({x}\right), \ldots, q_n \left({x}\right)}\right\rangle$ be differentiable vector-valued functions. The derivative of their dot product is given by: :$\dfrac \d {\d x} \left({\mathbf r \left({x}\right) \cdot \mathbf q \left({x}\right)}\right) = \mathbf r' \left({x}\right) \cdot \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \cdot \mathbf q' \left({x}\right)$
Borel Sigma-Algebra of Subset is Trace Sigma-Algebra
Let $\left({X, \tau}\right)$ be a topological space, and let $A \subseteq X$ be a subset of $X$. Let $\tau_A$ be the subspace topology on $A$. Then the following equality of $\sigma$-algebras on $A$ holds: :$\mathcal B \left({A, \tau_A}\right) = \mathcal B \left({X, \tau}\right)_A$ where $\mathcal B$ signifies Borel $\sigma$-algebra, and $\mathcal B \left({X, \tau}\right)_A$ signifies trace $\sigma$-algebra.
Derivative of Vector Cross Product of Vector-Valued Functions
Let: :$\mathbf r: t \mapsto \begin {bmatrix} x \\ y \\ z \end{bmatrix}$ :$\mathbf q: t \mapsto \begin {bmatrix} \chi \\ \gamma \\ \zeta \end{bmatrix}$ be differentiable vector-valued functions, where: :$x, y, z, \chi, \gamma, \zeta$ are (images of) differentiable real functions. The derivative of the vector cross product of $\mathbf r$ and $\mathbf q$ is given by: :$D_t \left({\mathbf r \left({t}\right) \times \mathbf q \left({t}\right)}\right) = \mathbf r' \left({x}\right) \times \mathbf q \left({x}\right) + \mathbf r \left({x}\right) \times \mathbf q'\left({x}\right)$
Existence and Uniqueness of Monotone Class Generated by Collection of Subsets
Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\mathfrak m \left({\mathcal G}\right)$, the monotone class generated by $\mathcal G$, exists and is unique.
Derivative of Product of Real Function and Vector-Valued Function
Let: :$\mathbf r:x \mapsto \mathbf z$ be a differentiable vector-valued function, where: :$\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \\ \vdots \\ z_n \end{bmatrix}$ such that: :$z_1, z_2, \cdots, z_n$ are (images of) differentiable real functions. Let: :$f: x \mapsto y$ be a differentiable real function. Then: :$D_x \left({y \, \mathbf z}\right) = \dfrac {\d y} {\d x} \mathbf z + y \dfrac {\d \mathbf z} {\d x}$
Generated Sigma-Algebra by Generated Monotone Class
Let $X$ be a set, and let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a nonempty collection of subsets of $X$. Suppose that $\mathcal G$ satisfies the following condition: :$(1):\quad A \in \mathcal G \implies \complement_X \left({A}\right) \in \mathcal G$ that is, $\mathcal G$ is closed under complement in $X$. Then: :$\mathfrak m \left({\mathcal G}\right) = \sigma \left({\mathcal G}\right)$ where $\mathfrak m$ denotes generated monotone class, and $\sigma$ denotes generated $\sigma$-algebra.
Heine-Borel Theorem/Metric Space
A metric space is compact  it is both complete and totally bounded.
Borel Sigma-Algebra on Euclidean Space by Monotone Class
Let $\left({\R^n, \tau}\right)$ be the $n$-dimensional Euclidean space. Then: :$\mathcal B \left({\R^n, \tau}\right) = \mathfrak m \left({\tau}\right)$ where $\mathcal B$ denotes Borel $\sigma$-algebra, and $\mathfrak m$ denotes generated monotone class.
Measure is Strongly Additive
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is strongly additive, that is: : $\forall E, F \in \Sigma: \mu \left({E \cap F}\right) + \mu \left({E \cup F}\right) = \mu \left({E}\right) + \mu \left({F}\right)$
Measure is Subadditive
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is subadditive, that is: :$\forall E, F \in \Sigma: \mu \left({E \cup F}\right) \le \mu \left({E}\right) + \mu \left({F}\right)$
Representation of Degree One is Irreducible
Let $\left({G, \cdot}\right)$ be a finite group. Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation of $G$ on $V$ of degree $1$.  Then $\rho$ is an  irreducible linear representation.
Irreducible Representations of Abelian Group
Let $\left({G, \cdot}\right)$ be a finite abelian group. Let $V$ be a non-null vector space over an algebraically closed field $k$. Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a linear representation. Then $\rho$ is irreducible iff $\dim \left({V}\right) = 1$, where, $\dim$ denotes dimension.
Schur's Lemma (Representation Theory)
Let $\left({G, \cdot}\right)$ be a finite group. Let $V$ and $V'$ be two irreducible $G$-modules. Let $f: V \to V'$ be a homomorphism of $G$-modules. Then either: :$f \left({v}\right) = 0$ for all $v \in V$ or: : $f$ is an isomorphism.
Kernel is G-Module
Let $\struct {G, \cdot}$ be a group. Let $f: \struct {V, \phi} \to \struct {V', \mu}$ be a homomorphism of $G$-modules. Then its kernel $\map \ker f$ is a $G$-submodule of $V$.
Image is G-Module
Let $\left({G, \cdot}\right)$ be a group and let $f: \left({V, \phi}\right) \to \left({V', \mu}\right)$ be a homomorphism of $G$-modules. Then $\operatorname{Im} \left({f}\right)$ is a $G$-submodule of $V'$.
Set Intersection Preserves Subsets/Families of Sets
Let $I$ be an indexing set. Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$. Let: :$\forall \beta \in I: A_\beta \subseteq B_\beta$ Then: :$\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha \subseteq \bigcap_{\alpha \mathop \in I} B_\alpha$
Set Intersection Preserves Subsets
Let $A, B, S, T$ be sets. Then: :$A \subseteq B, \ S \subseteq T \implies A \cap S \subseteq B \cap T$
Null Space Contains Zero Vector
Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n: \mathbf A \mathbf x = \mathbf 0}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin {bmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\  \vdots &  \vdots & \ddots &  \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end{bmatrix}$  is a matrix in the matrix space $\map {\MM_\R} {m, n}$. Then the null space of $\mathbf A$ contains the zero vector: :$\mathbf 0 \in \map {\mathrm N} {\mathbf A}$ where: :$\mathbf 0 = \mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$
Commutative Linear Transformation is G-Module Homomorphism
Let $\rho: G \to \operatorname{GL} \left({V}\right)$ be a representation. Let $f: V \to V$ be a linear mapping. Let: :$\forall g \in G: \rho \left({g}\right) \circ f = f \circ \rho \left({g}\right)$ Then $f: V \to V$ is a $G$-module homomorphism. {{explain|between which $G$-modules? (probably $\left({G, \rho}\right) \to \left({G, \rho}\right)$, which makes it rather tautologous)}}
Characterization of Measures
Let $\struct {X, \Sigma}$ be a measurable space. Denote $\overline \R_{\ge 0}$ for the set of positive extended real numbers. A mapping $\mu: \Sigma \to \overline \R_{\ge 0}$ is a measure : :$(1):\quad \map \mu \O = 0$ :$(2):\quad \mu$ is finitely additive :$(3):\quad$ For every increasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$, if $E_n \uparrow E$, then: ::::$\map \mu E = \ds \lim_{n \mathop \to \infty} \map \mu {E_n}$ where $E_n \uparrow E$ denotes limit of increasing sequence of sets. Alternatively, and equivalently, $(3)$ may be replaced by either of: :$(3'):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow E$, then: ::::$\map \mu E = \ds \lim_{n \mathop \to \infty} \map \mu {E_n}$ :$(3''):\quad$ For every decreasing sequence $\sequence {E_n}_{n \mathop \in \N}$ in $\Sigma$ for which $\map \mu {E_1}$ is finite, if $E_n \downarrow \O$, then: ::::$\ds \lim_{n \mathop \to \infty} \map \mu {E_n} = 0$ where $E_n \downarrow E$ denotes limit of decreasing sequence of sets.
Measure is Countably Subadditive
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then $\mu$ is a countably subadditive function.
Dirac Measure is Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$. Then $\delta_x$ is a measure.
Dirac Measure is Probability Measure
Let $\left({X, \mathcal A}\right)$ be a measurable space. Let $x \in X$, and let $\delta_x$ be the Dirac measure at $x$. Then $\delta_x$ is a probability measure.
Definition:Co-Countable Measure
Let $X$ be an uncountable set. Let $\Sigma$ be the $\sigma$-algebra of countable sets on $X$. Then the '''co-countable measure (on $X$)''' is the measure defined by: :$\mu: \Sigma \to \overline{\R}, \ \mu \left({E}\right) := \begin{cases} 0 & : \text{if $E$ is countable}\\ 1 & : \text{if $E$ is co-countable}\end{cases}$ where: : $\overline{\R}$ denotes the extended real numbers : $E$ is co-countable iff $X \setminus E$ is countable.
Co-Countable Measure is Measure
Let $X$ be an uncountable set. Let $\Sigma$ be the $\sigma$-algebra of countable sets on $X$. Then the co-countable measure $\mu$ on $X$ is a measure.
Counting Measure is Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Then the counting measure $\left\vert{\cdot}\right\vert$ on $\left({X, \Sigma}\right)$ is a measure.
Trivial Vector Space iff Zero Dimension
Let $V$ be a vector space. Then $V = \left\{{\mathbf 0}\right\}$ iff $\dim \left({V}\right) = 0$, where $\dim$ signifies dimension.
G-Submodule Test
Let $\left({V, \phi}\right)$ be a $G$-module over a field $k$. Let $W$ be a vector subspace of $V$. Then $\left({W, \phi_W}\right)$, where $\phi_W: G \times W \to W$ is the restriction of $\phi$ to $G \times W$, is a $G$-submodule of $V$ iff $\phi \left({G, W}\right) \subseteq W$.
Composite of Continuous Mappings is Continuous/Point
Let $T_1, T_2, T_3$ be topological spaces. Let the mapping $f : T_1 \to T_2$ be continuous at $x$. Let the mapping $g : T_2 \to T_3$ be continuous at $x$ at $\map f x$. Then the composite mapping $g \circ f : T_1 \to T_3$ is continuous at $x$ at $x$.
Null Space Closed under Vector Addition
Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf A \mathbf x = \mathbf 0}$ be the null space of $\mathbf A$, where: :$\mathbf A_{m \times n} = \begin {bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$, $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ and $\mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$  are matrices :the column matrix $\mathbf x_{n \times 1}$ is interpreted as a vector in $\R^n$. Then $\map {\mathrm N} {\mathbf A}$ is closed under vector addition: :$\forall \mathbf v, \mathbf w \in \map {\mathrm N} {\mathbf A}: \mathbf v + \mathbf w \in \map {\mathrm N} {\mathbf A}$
Null Space Closed under Scalar Multiplication
Let: :$\map {\mathrm N} {\mathbf A} = \set {\mathbf x \in \R^n : \mathbf {A x} = \mathbf 0}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end {bmatrix}$,  $\mathbf x_{n \times 1} = \begin {bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end {bmatrix}$, $\mathbf 0_{m \times 1} = \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$ are matrices where each column is an element of a real vector space. Then $\map {\mathrm N} {\mathbf A}$ is closed under scalar multiplication: :$\forall \mathbf v \in \map {\mathrm N} {\mathbf A} ,\forall \lambda \in \R: \lambda \mathbf v \in \map {\mathrm N} {\mathbf A}$