The Lebesgue Measure of a Lipschitz Image of a Compact Subset of $\mathbb{R}^n$

Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure. Regard Euclidean space $\mathbb{R}^{n}$ as a real vector space, with the sum of points, the scalar multiple, and the difference $x-y$ of points; write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ with $d_{E}(x,y)=\lVert x-y\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square; compactness refers to the topology of the open sets of $(\mathbb{R}^{n},d_{E})$, a topology by The Open Subsets of a Metric Space Form a Topology. Let $\mathcal{B}(\mathbb{R}^{n})$ be the Borel $\sigma$-algebra and $\lambda_{n}$ Lebesgue measure on it. Powers with natural exponent are those of Natural Number Power of an Element of a Field, and $\sigma_{n}=\lVert(1,\dots,1)\rVert$ is the positive real number with $\sigma_{n}^{2}=n$ introduced in Uniform Grids on a Half-Open Box and Grid Hulls of a Compact Set in $\mathbb{R}^n$.

Let $K\subseteq\mathbb{R}^{n}$ be nonempty and compact in $\mathbb{R}^{n}$, let $L\in\mathbb{R}$ satisfy $0\le L$, and let $T:K\to\mathbb{R}^{n}$ be Lipschitz with constant $L$, the metrics on $K$ and on $\mathbb{R}^{n}$ being the Euclidean ones; that is,

$$\lVert T(x)-T(y)\rVert\le L\,\lVert x-y\rVert\qquad\text{for all }x,y\in K .$$

Write $T(K)=\{T(x):x\in K\}$. Then $T(K)$ is nonempty and compact in $\mathbb{R}^{n}$, both $K$ and $T(K)$ belong to $\mathcal{B}(\mathbb{R}^{n})$ and have finite $\lambda_{n}$-measure, and

$$\lambda_{n}\bigl(T(K)\bigr)\le\bigl(2\,\sigma_{n}\,L\bigr)^{n}\,\lambda_{n}(K).$$