A measure theory book I am reading, Measure Theory and Fine Properties of Functions, 2nd ed., pp. 34, seems to have a bit wonky definition for the one dimensional Lebesgue measure:
$$\mathcal{L}^1(A):=\inf\{\sum_{i=1}^\infty\text{diam}(C_i):A\subset\bigcup_{i=1}^\infty C_i,C_i\subset \mathbb{R}\}$$
The authors do not seem to impose any restriction for the sets $C_i\subset\mathbb{R}$ while the definitions of the Lebesgue measure in $\mathbb{R}$ I have seen so far use e.g. closed/open intervals of $\mathbb{R}$. This raises the question that if we are in a metric space $(X,d)$ and define
$$m(A):=\inf\{\sum_{i=1}^\infty\text{diam}(C_i):A\subset\bigcup_{i=1}^\infty C_i,C_i\subset X\}$$
where the $C_i$s are allowed to be any subsets of $X$, are then all subsets of $X$ $m$-measurable?
Allowing for infinity as a possibility, the infimum always exists, so to that extent the measure $m(A)$ can always be defined. This is not the same as allowing all sets to be measurable -- the contradictions arise when you start imposing axioms on your measure space, e.g. additivity.
As for allowing arbitrary subsets $C_i$, this is an unusual approach but I don't think it changes anything. By allowing arbitrary $C_i$, it might seem that the Lebesgue measure can decrease, as you are taking the infimum over a larger set of covers. However, I claim that for any $\epsilon > 0$ you can replace an arbitrary cover with a cover using open/closed intervals and only increase the total sum of the diameters by $\epsilon$, so taking the limit $\epsilon \to 0$ you end up with the same infimum, and hence the same measure. Can you fill in the details here? (Hint: use an $\epsilon/2^n$ trick)