Understanding the concept of measure

Question

I am currently reading Lebesgue Integration on the Euclidean Space by Frank Jones , in my course of self study I came upon this doubt . The book defines the measure of an open set with the following definition , If $G\subset R^{n}$ is an open set and G is not an empty set , we define

$\lambda(G)$ = sup{$\lambda(P)| P\subset G$ , where P is a special polygon}

My doubt is whether the following proof of mine of a property that stems from the following definition is complete

To prove if $G_1 \subset G_2 $ then $\lambda(G_1) \leq \lambda(G_2) $

MY attempt

Let A = {$\lambda (P) | P \subset G_1 $ } B = {$\lambda(P)| P|\subset G_2 $ }

Now since $G_1 \subset G_2 $ we can say that $ A \subset B $ . Thus $sup(A) \leq sup(B)$

I ask this because the book mention the following lines extra in the proof for which I do not see any need Since $\lambda(G_2)$ is the least upper bound of the set B , it is an upper bound ofr the same set . There fore $\lambda(G_2)$ is also an upper bound for the smaller set A. As $\lambda(G_1)$ is the least upper bound of this set, we get $\lambda(G_1) \subset \lambda(G_2) $.

In a sense my question boils down to , the fact that I fail to understand why the explanation about the upper bounds supplied in the book is needed at all !

Answer 1

He's just saying the following in a convoluted way:

\begin{align} P\subset G_1 &\implies P \subset G_2 \\ &\implies \lambda (P) \leq \lambda (G_2) &&\text{definition of sup} \\\\ \therefore P \text{ arbitrary}&\implies \lambda(G_1) \leq \lambda(G_2) \end{align}

Answer 2

To conclude the proof, you need

Lemma: if $X \subseteq Y$ then $\mathrm{sup}(X) \leq \mathrm{sup}(Y)$

You are taking this as a given and using it without comment, and that's probably a fine thing to do.

The author however, decided that more explanation is needed, and repeated what is basically the main step in the proof of this lemma.