I was trying to work out the Lebesgue outer measure of a finite set.
I took a simple example, considered the finite set $\{a,b\}$.
Then, I wrote $\{a,b\}$ as $\{a\} \cup \{b\}$.
Since the Lebesgue outer measure of singleton sets is zero, therefore, it would probably turn out that the Lebesgue outer measure of the set $\{a,b\}$ that I chose is also zero as it is the union of two singleton sets.
To show this i assumed that if $\{a,b\}$ = $\{a\} \cup \{b\}$ then $m^*\{a,b\} \le m^*\{a\} + m^*\{b\}$, is there any property of lebesgue outer measure that says this ?
Also, I think that the same concept (of proving outer measure zero) can be extended to any countably finite or even infinite set?
Am I correct? I have just started learning this so I am just trying to get the basics right.
Best Answer
Notice that if $A$ is finite (or at most denumerable), we have $A = \{ a_1, \ldots, a_n, \ldots \}$, and hence $$ 0 \leq \mathfrak{m}^*A = {\frak m}^*\left( \bigcup_{n \geq 1} \{a_n\} \right) \leq \sum_{n \geq 1} {\frak m}^*\{a_n\} = 0$$ by subadditivity of $\frak m^*$, so ${\frak m}^*A = 0 $.