Families of surfaces lying in a null set

That is, a smooth one-parameter family of measurable hypersurfaces may be translated to lie in a null set. Moreover, the translations may be taken parallel to R and need not depend on g. Our motivation came from studying curved analogues of the Kakeya and Nikodym problems: we wished to show that a null set could contain a translate of every member of a specified family of curves, or in the Nikodym case, a curve from a specified family through every point. So we generalise Sawyer’s result in two ways. First, we will allow any codimension since curves of course have codimension n − 1. Second, we do not want to be restricted to using translations, since in the Nikodym case it is the “directions” or “shapes” we are allowed to vary while the positions are kept fixed. So we remove all distinction between “shape parameters” and “position parameters”, simply denoting those that are “given” by y and those we are free to choose by ω. Consider objects of the following form