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The Lebesgue inner measure is often not defined, for reasons explained here. Yet Tao, in his book An Introduction to Measure Theory, defines the Lebesgue inner measure for a bounded set $E$ as follows:

$$m_*(E) := m(A) - m^*(A\setminus E)$$

for any elementary set $A$ containing $E$, where $m^*$ is the Lebesgue outer measure.

Such section left me wondering if we could define the Lebesgue inner measure for an arbitrary set $E$ as follows:

$$m_*(E) = \sup\{ m(A) - m^*(A\setminus E) : \text{ $A$ is an elementary set }\}.$$

Would such a definition be 'correct'? In particular, are the following equivalent?

  • $E$ is Lebesgue measurable i.e. $E$ is Carathéodory measurable in $\mathcal{\mathbb{R}^n}$.
  • $m_*(E) = m^*(E)$.
Sam
  • 5,208
  • I guess if $E$ is Lebesgue measurable and $A$ is any bounded elementary set then both $A\setminus E$ and $A\cap E$ are Lebesgue measurable and $m(A)-m^(A\setminus E)=m^(A\cap E)$, so the supremum in the expression for $m_(E)$ equals $m^(E)$. – Alex Ravsky Oct 04 '24 at 20:48

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