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A BV space function is approximately differentiable at almost every point with respect to Lebesgue measure. Its approximate gradient is the density of the absolutely continuous part of its distributional derivative. This does not imply membership in a Sobolev space.

A Borel set of Lebesgue measure zero. Every Lebesgue-null set is contained in a Borel null set, by outer regularity; consequently testing all Borel null sets suffices in the definition of a Sierpiński set.

The distributional derivative of this indicator function is $e_1{\mathcal{H}}^1$ on the vertical segment $x_1=1/2$, so its total variation seminorm on a domain is one. This is singular with respect to planar Lebesgue measure, whereas Sobolev space weak derivatives in $W^{1,1}$ have integrable densities. Its bounded-variation space norm is $1/2+1=3/2$.

For a nonempty bounded subset $E$ of Euclidean space, let $N_{\delta }(E)$ be the least number of radius-$\delta$ balls covering $E$. Its lower and upper box-counting dimensions are the lower and upper limits of $\log N_{\delta }(E)/\log (1/\delta )$ as $\delta \downarrow 0$. If these limits agree, their common value is its Minkowski dimension. A cover by $N_{\delta }(E)$ balls implies $|E_{\delta }|{\lesssim }_n{\delta }^n{N}_{\delta }(E)$ for the open $\delta$-neighborhood. Conversely, a maximal separated subset gives the reverse comparison up to fixed changes of scale. This connects small-scale covering counts with neighborhood Lebesgue measure.

For probability measures on ${\mathbb{R}}^d$ with finite second moments and a source satisfying absolute continuity of measures with respect to Lebesgue measure, the quadratic Kantorovich optimal transport problem has a unique optimal transport plan. It is induced by the gradient of a convex function, which also uniquely solves the Monge optimal transport problem up to a source-null set.

The zero set $\{t\ge 0:B_t=0\}$ of one-dimensional Brownian motion is almost surely closed, uncountable, and has zero Lebesgue measure. The last fact follows from Tonelli theorem because $\mathbb{P}(B_t=0)=0$ for every $t>0$.

For a compact circle set with normalized Lebesgue measure at most $\exp (-8\pi M/\epsilon )$, where $M\ge 1$ and $0<\epsilon \le 1$, the displayed bounds can be realized by a trigonometric polynomial with spectrum in a positive-frequency interval arbitrarily far from zero. Take the logarithm of a positive-real-part Schwarz integral on the unit disk of a small-mean cutoff, truncate after radial dilation, and modulate its bounded imaginary part. A prefix isolates one analytic half and thus reveals the large real part. The separation between small function norm and large Fourier partial sums drives the Kahane-Katznelson divergence theorem.

Every subset of a Lebesgue-null set is Lebesgue measurable and has measure zero.

The covolume of a full Euclidean lattice is the Lebesgue measure of a fundamental parallelepiped in its ambient inner-product space. In orthonormal coordinates it is $|\det B|$ for a lattice basis matrix $B$. Scaling the lattice by $t>0$ multiplies the covolume by $t^n$; the dual lattice has reciprocal covolume.

Let $j(\mathfrak{b})$ be the Minkowski embedding of a number field applied to a nonzero fractional ideal. Under the metric $|x{|}^2={\sum }_{v\text{ real}}{x}_v^2+2{\sum }_{v\text{ complex}}|x_v{|}^2$, its covolume is $\sqrt{|D_K|}N(\mathfrak{b})$. For ordinary coordinate Lebesgue measure $\prod d{x}_v\prod d\mathrm{Re}{z}_{v}d\mathrm{Im}{z}_v$, its covolume is instead $2^{-r_2}\sqrt{|D_K|}N(\mathfrak{b})$. Here $D_K$ is the field discriminant, and the absolute norm of a fractional ideal is positive and multiplicative. The formula follows by taking the embedding determinant of an integral basis, then using the index of an integral ideal and scaling to handle a fractional ideal.

For nonnegative measurable $a$, define $(U_a(t,s)g)(x,v)=e^{-{\int }_s^{t}a(r,x-v(t-r),v)dr}g(x-v(t-s),v)$. The shear preserves Lebesgue measure and the multiplier lies in $[0,1]$, so this is a contraction on Lp spaces. Extended nonnegative integrals use $e^{-\infty }=0$. Characteristic local integrability gives the usual continuous trace at the starting time.

Every open set of finite Lebesgue measure in Euclidean space is covered, up to a null set, by countably many pairwise disjoint open balls contained in it. At each stage cover a compact subset of the open remainder, choose a packing with the Wiener covering lemma, and remove its closed balls. A fixed positive fraction of the remaining measure is removed each time, while sphere boundaries are null.

If $E\subseteq [0,1]$ is a Lebesgue measurable set and $0\le b\le \lambda (E)$, some measurable subset $B\subseteq E$ has Lebesgue measure $b$. The function $F(t)=\lambda (E\cap [0,t])$ has Lipschitz continuity with constant $1$, starts at $0$ and ends at $\lambda (E)$; the intermediate value theorem proves the assertion. Applying the assertion successively to remaining portions of $E$ gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most $\lambda (E)$. This is a concrete consequence of non-atomic measure structure, which fails for general measures containing an atom of a measure.

A sequence $(x_n)$ in $\mathbb{R}/\mathbb{Z}$ is equidistributed if the limiting frequency in every half-open interval equals that interval's length. Equivalently, its empirical probability measures converge to normalized Lebesgue measure when tested against every continuous function.

The integer multiplication map on the circle is an ergodic transformation for Lebesgue measure. In the Fourier basis, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.

A Euclidean body is a nonempty bounded measurable subset of ${\mathbb{R}}^n$ of positive Lebesgue measure. Projection inequalities are unchanged by modifying a body on a set of measure zero; geometric equality statements commonly impose additional regularity.

A weakly stationary process has a time-series spectral density precisely when its spectral measure of a stationary time series is absolutely continuous with respect to Lebesgue measure. Absolute summability of its autocovariances is sufficient and yields a continuous Fourier-series density. It is not necessary for existence; for a general integrable density, Fejér sums give an $L^1$ recovery formula.

Particles keep their velocity and move along straight trajectories. The method of characteristics gives $f(t,x,v)=f_0(x-tv,v)$. The shear preserves Lebesgue measure and every Lp norm. Compact initial support remains compact at each finite time, but need not remain in one compact set for all times.

Let $p,q\ge 2$ be relatively prime integers. If $\nu$ is invariant and ergodic for the integer multiplication map on the circle $T_p$ and $h_{\nu }(T_p)>0$, then for $\nu$-almost every $x$ the sequence $(T_q^{n}x)$ is an equidistributed sequence for Lebesgue measure. For a non-ergodic $T_p$ invariant measure, the same conclusion holds if almost every measure in its ergodic decomposition has positive entropy. Positive entropy of the whole measure alone does not suffice.

The Hamiltonian $(v^2-x^2)/2$ gives $\dot{x}=v$, $\dot{v}=x$. Its characteristic flow map is multiplication by $A_t$, with $A_t{A}_s=A_{t+s}$ and determinant one. The backward map is $A_{-t}$. This expanding-contracting linear flow preserves phase-space Lebesgue measure and every finite-$p$ integral of a transported density, despite stretching its level sets.