Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$?

Question

Suppose $m$ is Lebesgue measure. Define $x + A = \{x + y : y \in A\}$ and $cA = \{cy : y \in A\}$ for $x \in \mathbb{R}$ and $c$ a real number. Let $A$ be a Lebesgue measurable set. I have two questions.

Does $m(x + A) = m(A)$?
Does $m(cA) = |c|m(A)$?

Question 2 is a special case of this question about $\mathbb{R}^n$ — Caleb Stanford, Jul 20 '16 at 16:56

score 1 · Answer 1 · answered Jul 20 '16 at 15:09

1

The lebegue measure is invariant by translations so the answer is yes it is. Just writing the definition also the second follow.

answered Jul 20 '16 at 15:09

Giovanni Siclari

183

Lebesgue measure, do we have $m(x + A) = m(A)$, $m(cA) = |c|m(A)$?

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