{'fraction_non_alphanumeric': 0.03790489317711923, 'fraction_numerical': 0.0034458993797381117, 'mean_word_length': 4.671875, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the continuity of Haar measure on topological groupoids

It is shown that continuity of a family of invariant (Haar) measures on a topological groupoid G is equivalent to the continuity of the implied convolution product f * g for all pairs of functions / and g. An example is given of a groupoid which admits no (continuous) Haar measure. It results, therefore, that the usual C*-algebra associated with a Haar measure on G cannot, in general, be constructed. Some remarks are included concerning the construction of Haar measures on the holonomy groupoid of a foliated manifold. 1. Introduction. Several definitions of invariant, or Haar measure have been investigated in the literature for locally compact topological groupoids G, and chronologically they appeared in the following order. First, Westman (17) proposed a definition for locally trivial G in which positive Radon measures iix are assigned to the morphism sets G(x, y), one for each pair (x, y) of objects of G. These measures are subject to suitable invariance and smoothness conditions in the nature of (I) and (II) below, called respectively "left invariance" and "continuity". In (16) he gave a somewhat different definition for a principal groupoid; in this case positive Radon measures lix are given, indexed by the object set X of G, defined on G and supported by the sets Gx = \Jvx x e X) of positive Radon measures defined on G with supp(^x) = Gx for each x, and satisfying