open intervals with upper bound infinity are measurable

Proof

Suppose $A \subset R$ . Let $A_{1} = A \cap (a, \infty)$ and $A_{2} = A \cap (- \infty, a]$ . Now we want to show that $m^{*} (A_{1}) + m^{*} (A_{2}) \leq m^{*} (A)$ .

If $m^{*} (A)$ is infinite we are done. So suppose that $m^{*} (A) < \infty$ . Now, let ${I_{n}}$ be a collection of intervals such that

\sum_{n} ℓ (I_{n}) \leq m^{*} (A) + ε

And define

J_{n} = I_{n} \cap (a, \infty) K_{n} = I_{n} \cap (- \infty, a]

Then for each $n$ , each of $J_{n}, K_{n}$ are either an interval are empty and

$A_{1} \subset ⋃_{n} J_{n}$
$A_{2} \subset ⋃_{n} K_{n}$
$ℓ (I_{n}) = ℓ (J_{n}) + ℓ (K_{n})$
Thus we have

\begin{aligned} m^{*} (A_{1}) + m^{*} (A_{2}) & \leq \sum_{n} m^{*} (J_{n}) + m^{*} (K_{n}) \\ = \sum_{n} ℓ (J_{n}) + ℓ (K_{n}) \\ = \sum_{n} ℓ (I_{n}) \\ \leq m^{*} (A) + ε \end{aligned}

Then take $ε \to 0$ and we have the desired result.

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open intervals with upper bound infinity are measurable

References

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