I am considering the Banach space $A$ with $\sup$-norm, which is the uniform closure of functions on a segment that are continuous but a finite collection of jump points, where they have limits from the left and from the right. It looks like any finite complex measure on the segment naturally generates a functional on $A$: on the dense set of piecewise continuous functions one can take the usual integral (if a jump point has a point mass, we may split this mass into two equal parts for the left and right parts of the function). Is it true that the space of measures is dual to $A$?
My own comment: I see that the answer is negative: fix a point of the segment and take a number $c$, consider the functional that assigns to a function whose left and right limits at this point are $a_+$ and $a_-$ the number $c\cdot a_+ + (1-c)a_-$. On continuous functions this functional does not depend on $c$.
I would like to extend my question as follows: describe the space dual to $A$.