zero set of analytic functions

Question

In this document, the following result is proved:

``Let $A(x)$ be a real analytic function of (an open connected domain $\mathcal{U}$) of $\mathbb{R}^n$. If $A$ is not identifically zero, then the zero set $F = \{x \in \mathcal{U}|A(x) = 0\}$ has zero measure."

I have several questions regarding the above result:

1. Are the assumptions for openness and connectedness necessary for the above result?

2. Is there a similar statement for complex analytic functions, and how about the necessity of the above assumptions there? (Where can I find a reference?)

3. The definition of ``zero measure" in this document seems not the same as the Lebesgue measure (as stated in (i) on page 3 of the document.)

   It defines that a set is measure zero if it can be contained by a union of open balls, while the sum of the power of the balls' radius can be arbitrarily small.

   What is the name of the measure used in the above definition? Just simply "measure"?

   (Since there is the Lebesgue measure, so I assume that the above measure also has its own name.)

4. Finally, is there a similar statement when the $x$ is infinite-dimensional?

Thanks for your clarification.