Mixing
Debunking a false proof
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Consider $E = mathbb{R} times {0}$ in $mathbb{R}^2$ . Apparaently this set (the x-axis) has measure zero. However, consider the following set $$O_n = big{mathbb{R} times big(-frac{1}{n}, frac{1}{n}big)big} $$ and $lim_{n to infty} m(O_n) = lim_{n to infty} frac{2}{n} times infty neq 0$ , where $m(*)$ is the Lebesgue measure. Therefore, we don't have $lim_{n to infty} m(O_n) = m(E)$ . Can you debunk my false proof of $lim_{n to infty} m(O_n) = m(E)$ ? Since $E$ is Lebesgue measurable, there is an open set $F$ such that $m(Fsetminus E) < epsilon$ . Find $n$ such that $O_n subset F$ and $m(O_n) leq m(E) + epsilon$ , because $O_n$ can be made arbitrarily close to $E$ . $epsilon$ being arbitrary implies $m(O_n) leq m(E)$ and $n to infty$ shows $lim_{n to infty} m(O_n) leq m(E)$
