# Why is an uncountable union of null sets not necessarily a null set?

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!

• $\bigcup_{x\in [0,1]}\{x\} = ?$ – amsmath Sep 8 '18 at 23:15
• Null is not the same as empty. – Daniel Mroz Sep 8 '18 at 23:18
• @DanielMroz sorry, I didn't know that, after you said that, I found this stackexchange post, and that clears up a lot... Ironic that the definition I didn't question was the one I didn't understand... haha – SmallFish Sep 8 '18 at 23:27
• @SmallFish Yes to all your questions! – amsmath Sep 8 '18 at 23:36
• If real intervals are to have measure (and they should; what else would measure theory be good for) and a set of a single point is to have no measure (it it should because a point is "that which has no part" i.e. is ... measureless) then real intervals are uncountable union of single points and therefore we must have this possibility. – fleablood Sep 8 '18 at 23:40

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.

• Thanks, I think this was the most complete answer, so I've marked it as the answer. I feel really dumb now for asking this haha... – SmallFish Sep 8 '18 at 23:41
• @SmallFish No worries at all --- I expect this answer will probably help others who have a similar question in the future. – user326210 Sep 8 '18 at 23:44

Take any set $A$ which is not a null set. Then $A=\bigcup_{a\in A}\{a\}$. Now, note that each $\{a\}$ is a null set.

• Should I mark this as an answer? This clearly answers the question as I asked it. However, I was, for all intents and purposes, confused on the difference between a null set and an empty set, and the comments cleared that up for me before I got to your answer...? – SmallFish Sep 8 '18 at 23:30
• My opinion is that you should either rewrite the question or accept the answer. As written, this question will be found by people who search for information about uncountable unions of null sets, and they should know that this is the answer. Or you can rewrite the question to be about the difference between a null set and the empty set, and then get, say, Daniel Mroz to write an answer that you can accept. (Although in that case, it might just get marked as a duplicate.) – Toby Bartels Sep 8 '18 at 23:34
• @SmallFish That's up to you. – José Carlos Santos Sep 8 '18 at 23:35