\[\newcommand{\FGrp}{F_{\mathbf{Grp}}} \newcommand{\Sg}{\mathsf{Sg}} \newcommand{\Hom}{\mathsf{Hom}} \newcommand{\hom}{\mathsf{Hom}} \newcommand{\epi}{\mathsf{Epi}} \newcommand{\aut}{\mathsf{Aut}} \newcommand{\mono}{\mathsf{Mono}} \newcommand{\Af}{\langle A, f \rangle} \newcommand{\dom}{\mathsf{dom}}\newcommand{\cod}{\mathsf{cod}} \newcommand{\ran}{\mathsf{ran}} \newcommand{\id}{\mathsf{id}} \newcommand{\Id}{\mathsf{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\Proj}{\mathsf{pr}} \newcommand{\Con}{\mathsf{Con}} \newcommand{\Clo}{\mathsf{Clo}}\newcommand{\Pol}{\mathsf{Pol}} \newcommand{\Op}{\mathsf{Op}} \newcommand{\Th}{\mathsf{Th}} \newcommand{\Mod}{\mathsf{Mod}} \newcommand{\src}{\mathsf{src}} \newcommand{\tar}{\mathsf{tar}} \newcommand{\eval}{\mathsf{eval}} \newcommand{\fork}{\mathsf{fork}}\newcommand{\Type}{\mathsf{Type}} \newcommand{\comp}{\circ} \newcommand{\tick}{\mathsf{tick}} \newcommand{\Time}{\mathsf{Time}}\newcommand{\Tree}{\mathsf{Tree}}\newcommand{\Term}{\mathsf{Term}} \newcommand{\Mod}{\mathsf{Mod}}\newcommand{\Th}{\mathsf{Th}} \newcommand{\inv}{\ ^{-1}}\]

symbols¶

Acronyms and abbreviations
Symbols

Acronyms and abbreviations ¶

AC: absolutely continuous
a.e.: almost everywhere
cod: codomain
dcpo: directed-cocomplete poset
dct: dominated convergence theorem
mct: monotone convergence theorem
dom: domain
em: law of the excluded middle
ω-cpo: ω-chain cocomplete poset
ran: range

Symbols ¶

The list below shows what to type (e.g., in the vscode IDE with lean extension) to produce some special characters.

Ā: A\bar (the closure of the set A)
B̄: B\bar (the closure of the set B)
𝔸: \BbbA
𝔹: \BbbB
ℂ: \BbbC (complex numbers)
ℕ: \BbbN (natural numbers)
ℝ: \BbbR (real numbers)
ℝ*: \BbbR* or \BbbR^\ast (extended real numbers, \(ℝ^∗ = ℝ ∪ \{-∞, ∞\}\))
𝕋: \BbbT
ℤ: \BbbZ (integers)
ℬ: \mscrB (\(ℬ(X) =\) Borel σ-algebra of \(X\))
L: \mscrL
𝒫: \mscrP (𝒫(X) = the power set of X)
𝔅: \mfrakB (\(𝔅(X,Y) =\) bounded linear transformations from \(X\) to \(Y\))
𝔐: \mfrakM (usually a σ-algebra)
α: \alpha (often used to denote an uncountable index)
β: \beta
γ: \gamma
Γ: \Gamma
δ: \delta (usually a small real number)
Δ: \Delta
ε: \epsilon (usually a small real number)
ι: \iota
κ: \kappa (usually a large cardinal number)
λ: \lambda (usually a measure)
Λ: \Lambda
ρ: \rho
σ: \sigma
Σ: \Sigma
∑: \sum
∏: \prod
Π: \Pi
π: \pi
φ: \varphi
ϕ: \phi
Φ: \Phi
χ: \chi (χₐ = the characteristic function of a)
Χ: \Chi
ψ: \psi
Ψ: \Psi
ω: \omega
Ω: \Omega
ℵ: \aleph
𝚤: \imath
𝚥: \jmath
ℓ: \ell
fₗ: f\_l
f̄: f\bar
f̂: f\hat
f̃: f\tilde
h₁: h\_1
h₂: h\_2, etc.
∩: \cap (intersection)
⋂: \bigcap (intersection)
∪: \cup (union)
⋃: \bigcup (union)
∧: \and or \wedge (infimum)
⋀: \bigwedge (infimum)
∨: \vee or \or (supremum)
⋁: \bigvee (supremum)
¬: \neg (negation)
∘: \o or \circ
×: \x or \times
∃: \exists
∀: \all or \forall
∈: \in
∋: \ni
∉: \notin
∌: \nni
≤: \leq
≥: \geq
⊆: \subseteq
⊇: \supseteq
⊂: \subset
⊃: \supset
≪: \ll (\(λ ≪ μ\) means “\(λ\) is absolutely continuous with respect to \(μ\)”)
≫: \gg
⋆: \star
∗: \ast
≈: \approx
∼: \sim
≡: \equiv
≅: \cong (isomorphism or congruence relation)
⟨: \langle
⟩: \rangle
←: \leftarrow
→: \to or \rightarrow
↑: \uparrow
↓: \downarrow
⟹: \Longrightarrow
⟺: \iff
↦: \mapsto
↠: \twoheadrightarrow (a surjective map)
↣: \rightarrowtail (an injective map)
⤖: \twoheadrightarrowtail (a bijective map)
∅: \emptyset
⊢: \vdash
⊨: \vDash
⫢: \vDdash
⊧: \models
⋈: \bowtie