Suppose I have a collection of multivalued functions $f:[0,2\pi]\rightarrow\mathbb{R^{3+}}$. It is also known that this space is a vector space.
We define distance between two multivalued function $f^{(1)}$ and $f^{(2)}$ as: $\int_0^{2\pi}\sum_{i=1}^3|f^{(1)} _i(\theta)-f^{(2)} _i(\theta)|d\theta$.
Now, my question is: how to define probability measure on this space of multivalued function? Any idea is this regards is welcome.
Thank you.