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I am trying to show that lebesgue inner measure is translation invariant for bounded set. The lebesgue inner measure is defined as follows for bounded set:

$m_*(A)=|I|-m^*(I-A)$, where $m^*(A)$ is the lebesgue outer measure for $A$, $A \subset I$.

I know that lebesgue outer measure is translation invariant.

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See this post, Inner and outer Lebesgue measure and the PDF linked there, http://faculty.etsu.edu/gardnerr/talks/Measure-Theory.pdf. From those, we know that if $E$ is Lebesgue outer measurable, then $m_*(E) = m^*(E)$. Suppose $E$ is measurable. Then for $h \in \mathbb{R}$, $E + h$ is outer measurable, and \begin{align*} m_*(E+h) = m^*(E+h) = m^*(E) = m_*(E) \end{align*} Thus $m_*$ is translation invariant.

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    There is no need here for $E$ to be measurable, though – 2017-02-01
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    @Omnomnomnom You're right, I'm (unfortunately) using an extra hypothesis. You should be able to prove this without assuming $E$ is outer measureable. – 2017-02-01