Differentiability Almost Everywhere

1. In this paper we will give necessary and sufficient conditions for a measurable function to be equivalent to one which is differentiable a. e. on a set E. The condition is in terms of Marcinkiewicz type integrals which have also been recently the main objects in problems of differentiability. The reader is especially referred to the important paper on differentiability by E. M. Stein and A. Zygmund [8]. Their results are crucial for our paper; in fact, our paper is essentially only a slight refinement of their results and of a theorem due to Marcinkiewicz [2]. Let Ig — [0, l], and let/: I0—*R he a measurable function, where R is the set of real numbers. We will abbreviate the second symmetric difference of / at x by A2f(x, t), i.e., A2f(x, t) =f(x+t) +f(x-t)-2f(x). It will also be useful to retain the notation ex(t) = | A2f(x, t) | (2i)_1 introduced in [8]. We say that two functions are equivalent if they differ on a set of measure zero. The measure of a set A will be denoted by IAI. The function d>: R-+R is defined by (x) =0, \x\ ^ 1. We are now ready to state the main theorem of our paper.