Is there a concept of a 'measurable manifold'?

I just began learning about manifolds reading Lee's smooth manifolds. The notion of (Borel)-measurable functions generalizes continuity and at least integration in Lebesgue-Sense could still be done even if the domain is not a topological manifold.

I am thinking of a definition like: Let $(M,T)$ be a topological space. Let $\mathcal{A}=\sigma(T)$ be the $\sigma$-algebra induced by the topology $T$, i.e. the smallest $\sigma$-algebra that contains $T$. Then $(M,\mathcal{A})$ is called a measurable manifold, iff $T$ is hausdorff and second-countable (is this necessary?) and $\forall p \in M$ there is a open (wrt $T$) nbhd $U$, and a measurable bijection $\varphi:U \to \mathbb{R}^n$, where $\mathbb{R}^n$ is to be understood with Lebesgue-$\sigma$-algebra.

I'm just wondering, why I never came across the concept of a measurable manifold as generalization of a topological manifold.

One possible explanation in my mind is, that such a manifold would not have a well-defined dimension anymore, as I believe that there are measurable bijections between $\mathbb{R}$ and $\mathbb{R}^2$.

Would such a theory simply be to wild or boring or after all not too different from topological manifolds?