Why do we need Lebesgue to claim that the length of $\mathbb Q$ is $0$? Since we know that $\mathbb Q=\bigcup_{n\in\mathbb N}\{x_n\}$ (i.e. it's countable), why do we need Lebesgue to say that $\mathbb Q$ has length $0$ ? Indeed, if $$\operatorname{Length}(\mathbb Q)=\sum_{k\in\mathbb N}\operatorname{Length}(\{x_i\})=0,$$
since $\operatorname{Length}\{x_i\}=0$. So if this is not valid, how should the length of a set be defined?
 A: This is a great question - basically, you're starting the thought process which will ultimately lead to the full theory of Lebesgue measure/integration. You're right that that full theory isn't needed for simple examples, but those simple examples are not as simple as they first appear, and the full theory does develop naturally out of the analysis of those simple examples we're forced to do.

There are a couple points here (and, following you, I'm going to use "length" to describe the concept we're trying to pin down):


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*First of all, the claim $$\mbox{"The length of $\mathbb{Q}$ is $0$ since $Length(\mathbb{Q})=\sum_{n\in\mathbb{N}}Length(\{x_n\})=\sum_{n\in\mathbb{N}}0=0$"}$$ is extremely problematic - consider the "clearly false" claim $$\mbox{"The length of $\mathbb{R}$ is $0$ since $Length(\mathbb{R})=\sum_{r\in\mathbb{R}}Length(\{r\})=\sum_{r\in\mathbb{R}}0=0$."}$$ The difference between these two claims - the first of which seems right, the second of which seems stupid - is the additivity assumptions they're each making: the first says that length is countably additive, while the second says that length is continuum additive. So this reveals that we need to limit additivity - and hence think about it in some detail - if we're going to get a non-silly notion of length.


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*Actually, even countable additivity is problematic, as seen by the construction of Vitali sets. But that's a more subtle issue; I just want to emphasize that although your claim is extremely intuitive, it's hiding some real mathematical assumptions.


*More technically, the idea that we can satisfyingly set $Length(A)=\int_A1_A(x)dx$ using a Riemann integral is not going to work here - the indicator function of the rationals isn't Riemann integrable! So while it should certainly be true that $Length(A)=\int_A1_A(x)dx_{Riemann}$ when the latter exists, that's not going to serve us as a general definition.
So the point is that $(i)$ our existing notion of integration isn't up to the task and $(ii)$ we have genuinely interesting technical issues (additivity) we need to resolve. Now we do not need the full theory of Lebesgue measure/integration here - the rules "the length of a single point is $0$" and "length is countably additive" already imply $Length(\mathbb{Q})=0$ - but nobody ever claimed we do (to my knowledge); rather, Lebesgue measure/integration is the natural ending point of this line of thought.
(Well, "ending point" is a bit strong - but it represents a coherent, fairly complete technical idea which ties all these issues together and gets new interesting results.)
A: What you are saying is that Lebesgue's definition of measure (or length, if you insist) is obvious. Yes, it is, in retrospect! But Lebesgue was the first to define it explicitly.
