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By Tao T.

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Then we can write E as the finite union Q1 ∪ . . ∪ Qk of disjoint boxes, which need not be closed. But, similarly to before, we can use the epsilon of room strategy: for every ε > 0 and every 1 ≤ j ≤ k, one can find a closed sub-box Qj of Qj such that |Qj | ≥ |Qj | − ε/k (say); then E contains the finite union of Q1 ∪ . . ∪ Qk disjoint closed boxes, which is a closed elementary set. By the previous discussion and the finite additivity of elementary measure, we have m∗ (Q1 ∪ . . ∪ Qk ) = m(Q1 ∪ .

Show that the following are equivalent: (i) E is Lebesgue measurable with finite measure. (ii) (Outer approximation by open) For every ε > 0, one can contain E in an open set U of finite measure with m∗ (U \E) ≤ ε. (iii) (Almost open bounded) E differs from a bounded open set by a set of arbitrarily small Lebesgue outer measure. ) (iv) (Inner approximation by compact) For every ε > 0, one can find a compact set F contained in E with m∗ (E\F ) ≤ ε. (v) (Almost compact) E differs from a compact set by a set of arbitrarily small Lebesgue outer measure.

Iii) (Almost open bounded) E differs from a bounded open set by a set of arbitrarily small Lebesgue outer measure. ) (iv) (Inner approximation by compact) For every ε > 0, one can find a compact set F contained in E with m∗ (E\F ) ≤ ε. (v) (Almost compact) E differs from a compact set by a set of arbitrarily small Lebesgue outer measure. (vi) (Almost bounded measurable) E differs from a bounded Lebesgue measurable set by a set of arbitrarily small Lebesgue outer measure. (vii) (Almost finite measure) E differs from a Lebesgue measurable set with finite measure by a set of arbitrarily small Lebesgue outer measure.

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An introduction to measure theory by Tao T.


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