Examples of measurable spaces and measurable functions other than $(\mathbb{R}^n, \mathcal{L}) \to (\mathbb{R}_{+}, \mathcal{B})$.

Question

I'm studying Lebesgue integration theory. In this theory, we study morphisms from $(\mathbb{R}^n, \mathcal{L})$ to $(\mathbb{R}_{+}, \mathcal{B})$.

Are there other interesting or important examples of morphism in the category of measurable spaces and measurable functions?