Analysis with Programming

More proving, still on Real Analysis. This is my solution and if you find any errors, do let me know. Problems Lebesgue Measure : Let $\mu$ be set function defined for all set in $\sigma$-algebra $\mathscr{F}$ with values in $[0,\infty]$. Assume $\mu$ is countably additive over countable disjoint collections of sets in $\mathscr{F}$. Prove that if $A$ and $B$ are two sets in $\mathscr{F}$, with $A\subseteq B$, then $\mu(A)\leq \mu(B)$. This property is called monotonicity . Prove that if there is a set $A$ in the collection $\mathscr{F}$ for which $\mu(A)<\infty$, then $\mu(\emptyset)=0$. Let $\{E_{k}\}_{k=1}^{\infty}$ be a countable collection of sets in $\mathscr{F}$. Prove that $\mu\left(\displaystyle\bigcup_{k=1}^{\infty}E_{k}\right)\leq \displaystyle\sum_{k=1}^{\infty}\mu(E_k)$ Lebesgue Outer Measure : By using property of outer measure, prove that the interval $[0,1]$ is not countable. Let $A$ be the set of irrational numbers in the interval $[0,1]$. Prove that $...

Another proving problem, this time on Real Analysis. Problem Prove that the Lebesgue outer measure is translation invariant. (Use the property that, the length of an interval $l$ is translation invariant.) Solution Proof . The outer measure is translation invariant if for $y\in \mathbb{R}$, \begin{equation}\nonumber \mu^{*}(A)=\mu^{*}(A+y) \end{equation} Hence, we need to show that Case 1: $\mu^{*}(A)\leq \mu^{*}(A+y)$; and Case 2: $\mu^{*}(A+y)\leq \mu^{*}(A)$. Case 1 : Consider a countable collection $\{I_n\}_{n=1}^{\infty}$, and let \begin{equation}\nonumber W = \left\{\displaystyle\sum_{n=1}^{\infty}l(I_n)\mid A\subseteq\displaystyle\bigcup_{n=1}^{\infty}I_n\right\} \end{equation} Then the outer measure of $A$ is, \begin{equation}\nonumber \mu^{*}(A)=\inf\,\{W\}. \end{equation}