Crofton's intersection formula states that the $(n-j)$'th intrinsic volume of a compact convex set in $\mathbb{R}^n$ can be obtained as an invariant integral of the $(k-j)$'th intrinsic volume of sections with $k$-planes. This paper discusses the question if the $(k-j)$'th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique.
The answer is negative: we show that the sums of the $(k-j)$'th intrinsic volume and certain translation invariant continuous valuations of homogeneity degree $k$ yield counterexamples. If the measurement function is local, these turn out to be the only examples when $k=1$ or when $k=2$ and we restrict considerations to even measurement functions. Additional examples of local functionals can be constructed when $k \geq 2$.
Keywords: Crofton's formula, Klain functional, Local functions, Spherical lifting, Uniqueness, Valuation