If we measure a number, we get an interval. What if we measure a function or an operator?

Assume that we measure a physical ~mntity x with a measuring device whose accuracy is ~ (i.e., whose prtniucers guarantee that the difference a: ~ between the actual value :r and the measured value d{ms m~t exceed 5). If the result of this measurement is ~, then Ix~ssible values of a: fiwm an interval [~-~,~+~]. Suppose nnw that we know that a physical qtiantity y is a function of the physical quantity :2: (in other words, we know that V = f ( x ) for .~,me ftinction f ( z ) ) , but we do not know f . How to determine f ? We can nmasure only finitely many values, with finite precisitm, .~, alter finitely ninny measurements, we get a set of possible functions f ( x ) . This set can be called a fiow21on inten*td (function intervals were first analyzed by R. Mt~re himself). T h e situation can bectnne even more ceanplicated.. For example, if we analyze how phyfical fields evolve, then in addititm to functitms, we must d e . ~ h e tWrators, i.e., mappings that transfi~rm a functitm (current value f(~') of a physical field) into a function (predicted futtire value of this field). Again, since we can perfi~rm only finitely many measurements, at any moment of time, tmr meastirement results are consistem with the whole hunch of different operators. So, at any moment of time, we have a .set of operators; we can call it an ~urrator imerwd. One can apply different ideas to describe f imaion intervals, operator intervals, etc. But it is desirable to develop a general fi~mmlism that would cover all these cams. In this paper, we proI~se and justify such a fi~rmalism.