Lebesgue-Measure on Euclidean Space and Natural way to extend Lebesgue measure on normed space

I will ask two questions that involve proposition 1.15 and theorem 1.16 from Follands' real analysis book.

First question:

Folland constructs the Lebesbgue Stieltjes measure in $\mathbb{R}$, and I already know that it is possible to construct the Lebesbgue Stieltjes measure in $\mathbb{R}ˆn$ by extending the statements of proposition 1.15 and theorem 1.16 in the most natural. My question is the following:

Is it possible to construct the Lebesgue Stielsjes measure on $\mathbb{R}ˆn$ starting from the Lebesgue Stieltjes measure on $\mathbb{R}$ and somehow extending it to the sigma algebra product?

Second question:An arbitrary normed space is locally connected and locally convex. So if E is a normed space, then:

• Every point of $E$ has a local basis composed of convex sets.
• If $A$ is open in $E$, then there exists a unique partition of $A$ constituted by open connected sets that are closed in $A$.

Suppose that $E$ is a normed space such that:

• 1° If $A$ is open in $E$, then any convex subset of $A$ that is open in $E$ is closed in $A$
• 2° every disjoint collection of open subsets of $E$ in $E$ is countable.

(I don't know if the 1° property is always valid or if it is valid by adding a specific topological property as a condition). The 2° property is valid if $E$ has a countable dense subset (for exemple, if $E$ is Lindelöf)

Since convex implies connectedness, then if $E$ has such properties, then we conclude that:

• if $A$ is open in $E$, then there exists a unique partition of $A$ that consists of open covex sets that are closed in $A$, and this partition is countable.

Thus, if $E$ has this property, we can probably extend the concept of Lebesgue measure (not lebesgue-stielsjtes, due to monotonicity) to $E$, since this is the property that Folland uses to prove proposition 1.15 to construct the Lebesgue measure.

I think that if we put $\mu_0(\bigcup(u_j,v_j)) = \sum(||u_j-v_j||)$, (where $(u_j,v_j)$ is the segment of extrems $u_j$ and $v_j$) on proposition 1.15 we can get an natural extension of the lebesgue measure on $E$. And probably this measure would have good properties such the Lebesgue measure on \mathbb{R} has, such invariance under homotety and translation.

I'll be grateful if someone can help me formalize all this (I'm studying measure theory for the first time and have only read the first chapter of Folland's book), and perhaps find a good condition for a normed space $E$ own the 1° property.