Lebesgue measure and similarities There is a well-known theorem in Euclidean geometry (Eucl. VI-19) that says that _the ratio of the areas of any two similar polygons is equal to the square of the corresponding ratio of similarity_. In many texts treating the notion of area in the sense of Jordan one can find the analogue of the above result for Jordan measurable subsets of the plane; the theorem in question goes like this: _Suppose_ $R$ _and_ $S$ _are Jordan measurable sets such that_ $R$ _is similar to_ $S$ _under a similarity with dilation factor_ $\delta$*. Then,* $\mathrm{area}(R) = \delta^{2} \mathrm{area}(S)$*.* I wonder if there is an analogue of the above theorem for Lebesgue measurable subsets of the plane... Let me thank you in advance for your answers, bibliographical suggestions, etc.

Yes, this is true.

More generally, if $A$ is Lebesgue-measurable and $T$ is a linear mapping, then $\mu(T(A)) = \det(T)\mu(A)$. Also, if $T$ is a translation, then $\mu(T(A)) = \mu(A)$. This implies your statement since any dilation is a composite of a linear mapping and a translation.

See Theorem 2.20 of Rudin's _Real and Complex Analysis, 3rd Ed._