I ran across this statement "…for instance, an uncountable union of null sets need not be a null set (or even a measurable set)…" while looking through Terence Tao's blog site (See the first statement of #5). Since I'm taking a course in measure theory right now, I thought it might be relevant to understand why this is true, but I really don't know where to start. (In fact, I'm not entirely sure if this is relevant to measure theory, but Dr. Tao did mention that such a union might not even be a measurable set…).
Intuition told me it might be similar to why 0 $\cdot$ $\infty$ is indeterminate, but I understood that problem to be one of definitions. As such, the only thing I have been able to come up with is that I don't understand the definition(s) either "null set", "uncountable", or "union" precisely enough for me to grasp yet. I'm leaning towards not fully understanding the term "uncountable," as my understanding of transfinites and ordinals is sketchy at best.
I was wondering if anyone could define these terms; point me towards something to read, learn, or think about; or provide an example of an uncountable union of null sets not being a null set?
Edit: I didn't realize null set and the empty set were different things. Thanks to everyone for the examples and definitions!
Best Answer
A null set is a set of zero measure; more informally, it is a set of points that has no area. For example, all countable sets— including the rationals $\mathbb{Q}$, the natural numbers $\mathbb{N}$, the empty set $\varnothing$, and singleton sets like $\{0\}$—are null sets in $\mathbb{R}$ [under the Lebesgue measure].
Any countable union of null sets is still a null set, but an uncountable union might not be. For example:
$$[-1,1] = \bigcup_{x\in[-1,1]} \{x\}$$
The inverval on the left is not a null set; it has length 2. However, we can write it as an uncountable union of singleton sets that are each measure 0. Hence an uncountable union of null sets can yield a non-null set.