Journal of Applied Analysis Vol. 5, No. 2 (1999), pp. 223–238 KILLING RESIDUAL MEASURES O. ZINDULKA Received May 6, 1998 and, in revised form, October 26, 1998 Abstract. A σ-finite Borel measure in a topological space is called residual if each nowhere dense set has measure zero. We show that in various types of spaces without isolated points there are no residual measures. Among these spaces are e.g. σ-spaces, locally metrizable spaces, locally separable spaces, spaces that have a σ-point–finite π- base, submanifolds. If a topological space is provided with a σ-finite Borel measure, then two natural σ-ideals arise from the underlying notions of smallness — the ideal of meager sets and that of null–measure ones. If the former is included in the latter, i.e. if each meager set has measure zero, the measure is said to be residual. We ask about topological properties that forbid the existence of residual measures. As to my knowledge, the first result in this direction appeared in 1934 by Szpilrajn [21] who proved that a finite Borel measure in a separable metric space without isolated points was not residual. In [19] Szpilrajn’s result was generalized to an arbitrary finite Borel measure in a metric space without isolated points, which weight was bounded by the first real–valued measurable cardinal. In his famous book [20] Oxtoby presented this result and wrote that this “ ... theorem is perhaps the ultimate generalization of [Szpilrajn’s] Theorem” . 1991 Mathematics Subject Classification. Primary 28C15; Secondary 54E18, 54E35. Key words and phrases. Borel measure, residual measure, schismatic measure, schis- matic space, meager set, network, π-base. ISSN 1425-6908 c  Heldermann Verlag.