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Let $D>0$ be the diameter of the bounded set $S$, and choose $x_0\in S$. Then $S\subseteq B(x_0,D)$. It is enough to prove a volumetric covering bound for the unit ball.

Choose a maximal $1/4$-separated set $N$ in $B(0,1)$. The balls of radius $1/8$ centred at points of $N$ are disjoint and lie in $B(0,9/8)$. Comparing volumes givesMaximality means that the balls of radius $1/4$ centred at $N$ cover the unit ball.

After translating and scaling, at most $9^n$ balls of radius $D/4$ cover $S$. Assign each point of $S$ to one covering ball containing it. This gives at most $9^n$ disjoint pieces, each of diameter at most $D/2<D$. Thus the claim holds with the absolute constant $C=9$.

The Jacobian determinant of a differentiable characteristic flow map measures its change of volume. The variational equation and the Liouville formula for a fundamental matrix give the displayed formula for a complete smooth field $b$. For kinetic advection with $b=(v,F)$, its divergence is ${\nabla }_v\cdot F$. A zero divergence preserves phase-space Lebesgue measure.

A probability measure on the circle invariant under $T_2$ and $T_3$ and ergodic for their jointly generated semigroup is Lebesgue measure if either transformation has positive Kolmogorov-Sinai entropy. Without joint ergodicity, a mixture of Lebesgue measure and an atomic invariant measure shows why positive entropy alone is insufficient.

The measure $d\sigma /\sigma$ is invariant under multiplication by every positive constant. In the logarithmic coordinate it is Lebesgue measure. It cannot be normalized over $(0,\infty )$, although a bounded truncation is a log-uniform distribution.

An uncountable subset $S$ of the real numbers whose intersection with every null set for Lebesgue measure is countable. Testing only Borel sets suffices because every Lebesgue-null set has a Borel null superset. No measurability of $S$ is assumed.

For $0<\kappa <8$, the probability that chordal ${\mathrm{SLE}}_{\kappa }$ approaches an interior point $z$ to conformal radius at most $ϵ$ is, on compact subsets of $\mathbb{H}$, comparable towhere $G_{\kappa }$ is the SLE Green function. The positive exponent implies that the trace has zero planar Lebesgue measure.

The spectral measure of a weakly stationary process is the finite nonnegative measure whose Fourier coefficients are its autocovariances. Its total mass is the process variance. A time-series spectral density exists exactly when this measure is absolutely continuous with respect to Lebesgue measure.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

Surface area is the two-dimensional measure induced on the unit sphere in three dimensions. It is invariant under orthogonal rotations. It is not the restriction of ambient three-dimensional Lebesgue measure, under which the sphere has measure zero. Normalizing by the sphere area gives a probability measure.

The full tent map on the unit interval isIts two affine branches expand lengths by two, and it preserves Lebesgue measure.

For $f\in L^1(\mathbb{T},m)$ with $m$ normalized Lebesgue measure, translation on the circle group is continuous in the L1 norm: $\Vert f(\cdot +t)-f{\Vert }_1\to 0$ as $t\to 0$ modulo one. Approximate $f$ in the L1 norm by a continuous function, use its uniform continuity, and use translation invariance to preserve the approximation error. In particular, $m(B△(B-t))\to 0$ for every measurable $B$.

Translation shifts the argument without changing the shape of a function. On the real line, the displayed convention shifts a graph to the right by $h$. Lebesgue measure invariance preserves every finite Lp norm; its Fourier transform is multiplied by $e^{-ih\xi }$.

For an irrational skew shift, the averages of every continuous function $g$ converge uniformly in the starting point to $\int gd{m}_2$, with $m_2$ normalized Lebesgue measure. Nonconstant Fourier basis characters have either linear or quadratic phases. Linear phases are bounded geometric series; for quadratic phases the Van der Corput inequality for finite scalar sequences reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by trigonometric polynomials proves the assertion and identifies every invariant Borel probability measure as $m_2$.

A continuous self-map of a compact metric space is uniquely ergodic when it has exactly one invariant Borel probability measure. An irrational rotation of the circle is uniquely ergodic: invariance multiplies its nonzero-index Fourier coefficients by nontrivial phases, forcing those coefficients to vanish. The Stone-Weierstrass theorem then identifies the invariant measure as normalized Lebesgue measure.

Differentiating a smooth characteristic flow map in its initial point gives this linear equation, with $Y(s)=I$. The Liouville formula for a fundamental matrix then gives $\det Y(t)=\exp ({\int }_s^t\mathrm{div}b(r,X(r))dr)$. Incompressible Hamiltonian flows preserve Lebesgue measure because this divergence vanishes.

Volume measures the size of a three-dimensional region. In Euclidean space it is the Lebesgue measure $V={\int }_{\Omega }dxdydz$; a layer of thickness $h$ over a planar region has $V=\int hdA$.

A subset of the radius-$R$ Euclidean ball admits an $\epsilon$-metric net of size at most $(1+2R/\epsilon {)}^n$. Take a maximal separated set: its disjoint radius-$\epsilon /2$ Euclidean balls lie in the radius-$(R+\epsilon /2)$ ball. Comparing Lebesgue measures gives the estimate. In particular, the unit sphere admits a net of radius one with at most $3^n$ points.

A finite collection of Euclidean open balls has a disjoint subcollection whose concentric triples cover the original union. Greedily retain a largest remaining ball and discard all intersecting ones. Each discarded ball has radius no greater than the retained one and lies in its triple. Lebesgue measure consequently bounds the original union by $3^d$ times the sum of the retained volumes. This covering lemma is distinct from inverse-closedness results for the Wiener algebra.