I'm reading Rudin's Real and Complex Analysis and I'm puzzled about theorem 2.20, in which the Lebesgue measure is constructed via the Riesz representation theorem. Specifically, parts (c) and (e) of theorem 2.20 are on the translation-invariance of the Lebesgue measure, and how composition of the Lebesgue measure with a linear transformation is a scalar multiple of the Lebesgue measure, respectively.
Proving these properties requires that translations of Lebesgue measurable sets are Lebesgue measurable, as are linear transformations of Lebesgue measurable sets. One might be able to prove these facts by showing such operations are continuous, since continuous functions are Borel measurable, and the Lebesgue measure is regular. I have some doubts that this is the best approach, since Rudin proves that bounded linear transformations are continuous much later in Chapter 5. Are there other ways to prove that linear transformations of measurable sets is measurable that might reasonably be adopted by a student of Rudin who hasn't gotten to Chapter 5?