In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson's (1966) nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius (2008) and Russell (2008), which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these authors suggested. Unlike Arntzenius's proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell's proposal and does not retreat to mere finite additivity. * I thank Jeffrey Russell for his very valuable input to multiple drafts of the paper. I thank Philip Bricker for his helpful feedback on early drafts of the paper. Thanks to Cian Dorr for his encouraging comments. Thanks to Tobias Fritz for helpful discussions. Special thanks to two anonymous referees of Journal of Philosophical Logic for their scrupulous read, very helpful comments, and for pressing me on important details.

Weyl's tile argument purports to show that there are no natural distance functions in atomistic space that approximate Euclidean geometry. I advance a response to this argument that relies on a new account of distance in atomistic space, called the mixed account, according to which local distances are primitive and other distances are derived from them. Under this account, atomistic space can approximate Euclidean space (and continuous space in general) very well. To motivate this account as a genuine solution to Weyl's tile argument, I argue that this account is no less natural than the standard account of distance in continuous space. I also argue that the mixed account has distinctive advantages over Forrest's (1995) account in response to Weyl's tile argument, which can be considered as a restricted version of the mixed account. * I thank Philip Bricker and Jeffrey Russell for very helpful guidance, feedback, and discussions. I thank the audience at my talks based on this paper in Metaphysical Mayhem at Rutgers University in 2018, and in Philosophy of Logic, Mathematics, and Physics Graduate Conference at the University of Western Ontario in 2019. Among the audience, I especially thank Cian Dorr for his helpful feedback. I'd also like to thank a referee of Synthese for pressing me on the application of my account to relativistic settings, which helps clarify the relevance of the account.