Translation Invariance of the Lebesgue Outer Measure

If $E \in \mathcal P (\mathbb{R})$ then shifting all elements $e \in E$ by adding some $a \in \mathbb{R}$ gives us a new set $E + a$. We would like the Lebesgue outer measure of $E$ to equal the Lebesgue outer measure of $E + a$, that is, $m^*(E) = m^*(E + a)$. Fortunately this is true and is called the translation invariance property of the Lebesgue outer measure.

Theorem 1 (Translation Invariance of the Lebesgue Outer Measure): Let $E \in \mathcal P(\mathbb{R})$ and $a \in \mathbb{R}$. Then $m^*(E + a) = m^*(E)$.

Here, the notation "$E + a$" denotes the set given by $E + a = \{ e + a : e \in E \}$.

• Proof: Let $(I_n)_{n=1}^{\infty}$ be a sequence of open intervals that cover $E$. Then $(I_n + a)_{n=1}^{\infty}$ is a sequence of open intervals that cover $E + a$. Therefore:

(1)

\begin{align} \quad m^*(E + a) \leq \sum_{n=1}^{\infty} l(I_n + a) = \sum_{n=1}^{\infty} l(I_n) \end{align}

• So for every sequence of open intervals $(I_n)_{n=1}^{\infty}$ that cover $E$ we have that $\displaystyle{m^*(E + a) \leq \sum_{n=1}^{\infty} l(I_n)}$. Hence:

(2)

\begin{align} \quad m^*(E + a) \leq m^*(E) \quad (*) \end{align}

• Now let $(I_n)_{n=1}^{\infty}$ be a sequence of open intervals that cover $E + a$. Then $(I_n - a)_{n=1}^{\infty}$ is a sequence of open intervals that cover $E$. Therefore:

(3)

\begin{align} \quad m^*(E) \leq \sum_{n=1}^{\infty} l(I_n - a) = \sum_{n=1}^{\infty} l(I_n) \end{align}

• So for every sequence of open intervals $(I_n)_{n=1}^{\infty}$ that cover $E + a$ we have that $\displaystyle{m^*(E) \leq \sum_{n=1}^{\infty} l(I_n)}$. Hence:

(4)

\begin{align} \quad m^*(E) \leq m^*(E + a) \quad (**) \end{align}

• From $(*)$ and $(**)$ we conclude that $m^*(E) = m^*(E + a)$. $\blacksquare$