What I Mean by ‘Vegetarian’ Lebesgue Theory

Gabriel Stolzenberg

By ‘vegetarian’ Lebesgue theory, I mean Lebesgue theory without the measure, only the Lebesgue spaces, defined as completions in certain norms of certain spaces of continuous functions. Although doing without the measure may seem perverse, we will offer evidence that, for topics ranging from the definition of convolution to the Fourier Inversion Theorem and beyond (e.g., Weiner’s Tauberian theorem), the ubiquitous habit of unreflectively reaching for the measure sometimes obscures a simple, illuminating piece of mathematics. (But we are not suggesting that we dispense with the measure. That would be absurd, if only because, when we want to measure, we need a measure.)

For example, in Analysis II (221-222), Serge Lang compares the two approaches (with or without the measure) for the task of proving that multiplication by x is injective on L₁([0,1]). With the measure, this is obvious because the elements of this space can be represented as equivalence classes of functions defined almost everywhere on [0,1]. But, although Lang believes that, without this representation, we would, in effect, have to reconstruct it in this special case, it suffices to note, firstly, that multiplication by x is obviously injective on L₁([r,1]) for r > 0 (because, in that case, norm |xf| is bounded below by norm r|f|) and, secondly, that the L₁([r,1]) norm of any f in L₁([0,1]) is continuous at r = 0. (Proof. It follows directly from the measureless definition of L₁([r,1]) that, on [0,1], this function of r is a uniform limit of uniformly continuous functions and, hence, uniformly continuous.)

Thus, the two alternatives can be thought of in the following way. With the measure, the elements of a Lebesgue space can be represented as equivalence classes of functions for the relation, f = g almost everywhere. This is thought to be a good thing, except that the functions need only be defined almost everywhere and, hence, up to equivalence, nowhere—which is not thought to be a good thing. Without the measure, the elements of a Lebesgue space are not functions, which is thought to be a bad thing. But they are limits in the Lebesgue norms of nice functions, for example, ones obtained by smoothing, which is a very good thing indeed.

Updated February 26, 2005