In many practical applications, it turns out to be useful to use the notion of fuzzy transform: once we have non-negative functions A1(x), ..., An(x), with A1(x) + ... + An(x) = 1, we can then represent each function f(x) by the coefficients Fi which are defined as the ratio of two integrals: of f(x) * Ai(x) and of Ai(x). Once we know the coefficients Fi, we can (approximately) reconstruct the original function f(x) as F1 * A1(x) + ... + Fn * An(x). The original motivation for this transformation came from fuzzy modeling, but the transformation itself is a purely mathematical transformation. Thus, the empirical successes of this transformation suggest that this transformation can be also interpreted in more traditional (non-fuzzy) mathematics as well.
Such an interpretation is presented in this paper. Specifically, we show that fuzzy transform has a natural probabilistic interpretation -- related to the known interpretation of fuzzy sets as equivalence classes of random sets. We also show that a similar interpretation is possible for fuzzy control techniques.