Bijection and outer measure of two sets

Question

Suppose I have two sets and they have the same cardinality which I believe means that there exists a bijection between them, kind of like an isomorphism but I don't know if I can call it that since all sets may not be groups. Eg, let's consider the set $(0,1)$ and the set of irrationals between $(0,1)$, since they both have same cardinality, will they both have same outer measure ( which I know they have, but is this coincidence or actually due to bijection?)

Answer 1

In general, two sets with the same cardinality do not need to have the same outer measure. For example, all non-empty open intervals in $\mathbb{R}$ have the same cardinality, but intervals $(a, b)$ and $(c, d)$ have the same outer measure if and only if $b-a = d-c$.

Another important example is that the Cantor set has the same cardinality as $\mathbb{R}$, but has (outer) measure $0$.

In your specific example of $(0, 1)$ and $(0, 1) \setminus \mathbb{Q}$, these do have the same outer measure, but it isn't just because there is a bijection between them.