I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$.

The outer Lebesgue measure $E \subset \mathbb{R}^{n}$:

$$\lambda^{*}(E) = \inf\limits \{ ~\sum_{k=0}^{\infty} \text{vol}(Q_{k}):~\{Q_{k}\}_{k} \text{ are rectangles covering } E \text{ such that } E \subset \bigcup_{k=0}Q_{k} \} $$

The following properties are satisfied:

  • $\lambda^{*}(E) \geq 0 ~~~ \forall E \subset \mathbb{R}^{n}$
  • $\lambda^{*}(\emptyset) = \lambda^{*}(\{x\}) = 0 ~~\forall x \in \mathbb{R}^{n}$
  • If $E \subset F$ then, $\lambda^{*}(E) \leq \lambda^{*}(F)$


A subset $E$ of $\mathbb{R}^{n}$ is Lebesgue measurable if for every $F$ subset of $\mathbb{R}$, we have $$\lambda^{*}(F) = \lambda^{*}(E \cap F) + \lambda^{*}(F \setminus E)$$ Then we can let $\mathcal{M}$ denote the family of Lebesgue measurable subsets of $\mathbb{R}^{n}$

Would you recommend any changes to this and are there recommendations of books/notes where the Lebesgue measure in $\mathbb{R}^{n}$ is described in a good concise way?



You have the right ideas. Presumably, $F$ should be any subset of $\Bbb R^n$; what you're describing there is Carathéodory's criterion. Also, I think you have some extra words in you definition of Lebesgue measure; I would write $$ \lambda^{*}(E) = \inf\limits \{ ~\sum_{k=0}^{\infty} \text{vol}(Q_{k}):~\{Q_{k}\}_{k} \text{ are rectangles such that } E \subset \bigcup_{k=0}Q_{k} \} $$ Note that it doesn't make a difference if you define rectangles to be open or closed.

As for references: I would look through Stroock.


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