List of Measures

Introduction

This article lists the measures the corpus constructs and uses. A measure is a countably additive function on a $\sigma$-algebra of sets, and a row below names one measure, records the family of sets it measures and the property that distinguishes it — invariance, dimension, finiteness, atomicity — and points to the article that introduces it. Every row points to an article; this article introduces nothing and proves nothing. The measures are grouped by the layer that introduces them: the abstract measures and their constructions, the measures of Euclidean space and of its subsets, the invariant measures on groups, and the measures that arise in potential theory, probability and spectral theory. The non-examples — the set that carries no measure, the outer measure that is not a measure, the finitely additive function that is not countably additive — are recorded beside the measures as rows whose property cell names the failure.

Abstract Measures and Their Constructions

Object	The sets it measures, and the property it has	Introduced in
a measure $\mu$	the sets of a $\sigma$-algebra $\mathcal A$ on $X$; countably additive, $\mu(\emptyset) = 0$	Measure Theory and Integration
a probability measure	every measurable set, with $\mu(X) = 1$	Measure Theory and Integration; Measure-Theoretic Probability
a finite measure, a $\sigma$-finite measure	the measurable sets; $\mu(X) < \infty$, or $X$ a countable union of finite-measure sets	Measure Theory and Integration
a complete measure	the measurable sets and the subsets of null sets; every subset of a null set is measurable and null	Measure Theory and Integration
an outer measure $\mu^*$	every subset of $X$; monotone and countably subadditive, but not additive in general	Measure Theory and Integration
a premeasure on an algebra	the sets of an algebra $\mathcal A_0$; extends to a measure on $\sigma(\mathcal A_0)$ by Carathéodory	Measure Theory and Integration
the counting measure	every subset of a discrete set $X$, by cardinality; its atoms are the singletons	Measure Theory and Integration
the Dirac measure $\delta_x$	every measurable set, by $\delta_x(A) = 1$ if $x \in A$; the only atom is $\{x\}$	Measure Theory and Integration
a sum $\mu + \nu$ and a normalisation $\mu/\mu(X)$	the measurable sets; the normalisation is a probability measure	Measure Theory and Integration
an atom, and a diffuse measure	the measurable subsets of a set of positive measure; a diffuse measure has no atoms	Measure Theory and Integration

Measures on Euclidean Space

Object	The sets it measures, and the property it has	Introduced in
Lebesgue measure $\lambda^n$	the Borel and Lebesgue measurable sets of $\mathbb R^n$; translation-invariant, $\lambda^n([0,1]^n) = 1$	Measure Theory and Integration
the product measure $\mu \otimes \nu$	the sets of $\mathcal A \otimes \mathcal B$ on $X \times Y$; $(\mu \otimes \nu)(A \times B) = \mu(A)\nu(B)$	Measure Theory and Integration
the Hausdorff measure $\mathcal H^s$	every subset of a metric space, by covers of diameter at most $\delta$; it sees dimension	Geometric Measure Theory
Hausdorff dimension $\dim_{\mathcal H}$	every bounded set; the exponent at which $\mathcal H^s$ jumps from $+\infty$ to $0$	Geometric Measure Theory; Fractal Geometry
the Radon measure on a locally compact space	the Borel sets, finite on compact sets; produced by the Riesz representation theorem	Locally Compact Groups and Haar Measure; Distributions and Fundamental Solutions

Invariant Measures on Groups

Object	The sets it measures, and the property it has	Introduced in
the Haar measure $\mu_L$	the Borel sets of a locally compact group, finite on compacta; left-invariant, unique up to scale	Locally Compact Groups and Haar Measure
the right Haar measure $\mu_R$	the Borel sets; $d\mu_R = \Delta^{-1}d\mu_L$	Locally Compact Groups and Haar Measure
the modular function $\Delta$	the group, measuring the failure of bi-invariance; a continuous homomorphism to $\mathbb R_{>0}$	Locally Compact Groups and Haar Measure
a unimodular group	the Borel sets; abelian, compact, discrete, nilpotent and semisimple groups have a bi-invariant measure	Locally Compact Groups and Haar Measure
the normalised counting measure $\lvert G\rvert^{-1}\sum_g\delta_g$	every subset of a finite group; a bi-invariant probability measure	Measure Theory and Integration
the additive Haar measure on $\mathbb Q_p$	the balls $a + p^n\mathbb Z_p$, with $\mu(a+p^n\mathbb Z_p) = p^{-n}$ after $\mu(\mathbb Z_p) = 1$	p-adic Integration
the multiplicative Haar measure $d^\times x$	the Borel sets of $\mathbb Q_p^\times$; $\mu^\times(\mathbb Z_p^\times) = 1$	p-adic Integration
a quasi-invariant measure on $G/H$	the Borel sets of a homogeneous space, with Radon–Nikodym cocycle $\rho$	Locally Compact Groups and Haar Measure
the Tamagawa measure	the adelic quotient, assembled as the product of the local Haar measures	Adeles and Ideles; Adelic Analysis
an invariant measure of a dynamical system	the measurable sets of the phase space; preserved by the transformation	Ergodic Theory; Topological Dynamics

Measures in Potential Theory, Probability and Spectral Theory

Object	The sets it measures, and the property it has	Introduced in
the equilibrium measure	the compact subsets of $\mathbb R^n$; the measure of minimal energy, with capacity as its total mass	Potential Theory
the harmonic measure	the boundary of a domain; the distribution of the exit point of a Brownian path	Potential Theory; Brownian Motion and Stochastic Calculus
the Wiener measure	the Borel sets of the space of continuous paths; the law of Brownian motion	Brownian Motion and Stochastic Calculus
a Gaussian measure	the Borel sets of a Banach or Hilbert space; a centred Gaussian cylinder measure	Stochastic Partial Differential Equations
the law $\mu_X = \mathbb P \circ X^{-1}$ of a random variable	the Borel sets of the range; a probability measure	Measure-Theoretic Probability
a spectral measure $E(\lambda)$, $E$ projection-valued	the Borel sets of the spectrum; values are orthogonal projections	Banach and Hilbert Spaces; Operator Algebras
a signed measure $\nu$	the measurable sets, with values in $\mathbb R$; $\sigma$-additive and taking at most one infinite value	Measure Theory and Integration
a complex measure	the measurable sets, with values in $\mathbb C$	Measure Theory and Integration
the total variation $\lvert\nu\rvert$	the measurable sets; the least measure dominating $\nu$	Measure Theory and Integration
the Radon–Nikodym derivative $d\nu/d\mu$	the measurable sets; a density realising the absolutely continuous part of $\nu$	Measure Theory and Integration

Sets and Functions That Are Not Measures

Object	The failure	Introduced in
the Vitali set $V \subseteq [0,1)$	not Lebesgue measurable: no length can be assigned consistently with translation invariance and countable additivity	Measure Theory and Integration
an outer measure on all subsets	restricts to a measure only on the Carathéodory-measurable sets, and is not a measure on the whole power set	Measure Theory and Integration
the counting measure on an uncountable set	measures every set, but is not $\sigma$-finite	Measure Theory and Integration
a finitely additive set function	satisfies additivity over finite unions only, and need not be a measure	Measure Theory and Integration
a decreasing sequence without the finiteness hypothesis	continuity from above fails, as on $\mathbb N$ with counting measure and the sets $\{n, n+1,\dots\}$	Measure Theory and Integration

Summary

This list gathers the measures of the corpus: the abstract measures and their construction by Carathéodory, the counting and Dirac measures, Lebesgue and the Hausdorff measures of Euclidean space, the Haar measures of locally compact groups with the modular function and the adelic Tamagawa measure, and the measures of potential theory, probability and spectral theory. Each row records the family of sets the measure acts on and the property that separates it, and the final table records the constructions that carry the name of a measure and are not one.

Summary of Notation

The objects are named rather than denoted; the symbols occurring in the tables are collected here.

Symbol	Meaning
$\mu$, $\nu$, $\lambda$	measures; $\lambda^n$ Lebesgue measure on $\mathbb R^n$
$\mathcal A$, $\mathcal B$, $\mathcal B(X)$	a $\sigma$-algebra, the Borel $\sigma$-algebra
$\delta_x$	the Dirac measure at $x$
$\mathcal H^s$	Hausdorff $s$-measure
$\mu_L$, $\mu_R$, $\Delta$	left and right Haar measure, modular function
$d^\times x$	multiplicative Haar measure on $\mathbb Q_p^\times$
$\mu_X$, $\mu \otimes \nu$	law of $X$, product measure
$E(\lambda)$	spectral measure
$d\nu/d\mu$	Radon–Nikodym derivative