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The Lebesgue measure of a measurable set is the supremum of the measures of its compact subsets. In particular, an open set of finite positive measure contains a compact subset with at least half its measure. Together with a finite subcover, this turns a finite Wiener covering lemma estimate into an estimate for arbitrary open superlevel sets.

For an integer $K\ge 2$, multiplication by $K$ on the circle group preserves normalized Lebesgue measure. Each inverse image consists of $K$ branches, each scaling lengths by $1/K$.

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.

For $N\ge 2$, $U_{ij,\theta }F=F\circ R_{ij,\theta }$ uses the genuine two-coordinate rotation $(v_i,v_j)\mapsto (v_i\cos \theta +v_j\sin \theta ,-v_i\sin \theta +v_j\cos \theta )$. Average it uniformly over angles and pairs to obtain $Q$. Rotation invariance of Lebesgue measure makes $Q$ a bounded self-adjoint operator of norm one on $L^2({\mathbb{R}}^N)$. Each pair average is an orthogonal projection, but their full average is not in general a projection.

A bounded subset of ${\mathbb{R}}^n$ is a Kakeya set if it contains a unit line segment in every direction. Segment positions may vary with direction. Such a set need not have positive Lebesgue measure; the dimension problem concerns its Minkowski dimension or other notions of dimension.

A Kakeya tube is a neighborhood of transverse radius $\delta$ of a line segment of fixed unit length. Round cross-sections and comparable rectangular cross-sections are interchangeable up to fixed constants. The tube has Lebesgue measure comparable to ${\delta }^{n-1}$ in ${\mathbb{R}}^n$.

The doubling map preserves normalized Lebesgue measure on the unit circle. Each point has exactly two preimages, $x/2$ and $(x+1)/2$ modulo one, and each inverse branch scales lengths by $1/2$.

A Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to Lebesgue measure. In particular, it differs from a Borel set by a null set.

Such a measurable self-map preserves Lebesgue measure. On the unit interval examples include the identity, reflection $t\mapsto 1-t$, and $t\mapsto 2t(\mathrm{mod}1)$. Invertibility is not required.

Lebesgue outer measure on $\mathbb{R}$ iswhere the infimum is over countable interval covers. The Caratheodory measurability criterion selects the Lebesgue-measurable sets.

A function $f$ has Lusin's condition $(N)$ when it maps every Lebesgue-null set to a Lebesgue-null set.

For finitely many Lebesgue measurable sets $A_i\subseteq [0,1]$ and nonnegative demands $b_i$, pairwise disjoint measurable subsets $B_i\subseteq A_i$ with $\lambda (B_i)=b_i$ exist exactly when $\lambda ({\bigcup }_{i\in I}{A}_i)\ge {\sum }_{i\in I}{b}_i$ for every index subset $I$. Necessity is additivity and monotonicity of Lebesgue measure. For sufficiency, partition the set union into membership cells $E_J$, send flow from a source through demand vertices $i$ to cells with $i\in J$, then to a sink. Source capacities are $b_i$, cell capacities are $\lambda (E_J)$, and intermediate capacities are $D={\sum }_i{b}_i$. Every cut of a flow network has capacity at least $D$ by the assumed inequalities. The max-flow min-cut theorem provides the allocations, and divisibility of Lebesgue measure turns each cell allocation into disjoint pieces. If $D=0$, all $B_i$ can simply be empty.

For a nondecreasing real function $z$ on an interval $I$ and nonnegative measurable function $h$,The pushforward of $z^{\prime }(u)du$ is dominated by Lebesgue measure: on each interval, the integral of the derivative is at most the increase of $z$. This handles jumps and singular parts without a smoothness assumption.

Every circle rotation has the Furstenberg multiple recurrence theorem property for normalized Lebesgue measure. For a rational angle, use a period. For an irrational angle, choose a positive return time with angle close to zero; translation continuity in L1 on the circle then makes finitely many translates of a given positive-measure set simultaneously close to that set, and the union bound leaves a positive-measure intersection.

A measure on a topological space is outer regular on its measurable sets when the displayed open-cover identity holds. Lebesgue measure has this property. It allows a null set to be placed in arbitrarily small open neighborhoods, and a compact small set to receive a smooth cutoff of similarly small integral.

A partition atom is a member of a measurable partition. For a finite or countable partition, each cell is an atom of a sigma-algebra generated by that partition, whose measurable sets are unions of cells. A partition atom need not be an atom of a measure: an interval cell with positive Lebesgue measure can be split into smaller positive-measure sets that lie outside the generated partition sigma-algebra.