Estimates for the measure of noncompactness of bounded subsets of spaces of (Bochner-) integrable functions are obtained, a new class of condensing operators is discussed, and the solvability of a certain operator equation in a Hilbert space is proved. In this paper we discuss a new class of condensing operators, and we p r o ve t h e solvability of a certain operator equation. An extension of some results from 8] is obtained. Let us recall some deenitions. The measure o f n o n c ompactness

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... = E (U) 1] of a bounded set U in a normed space E is deened as the supremum of all numbersr > 0 such that there exists a sequence fu n g in U with ku n ; u m k k r for every n 6 = m. Given two B a n a c h spaces G and E, a continuous operator S : G ! E is called-condensing if E (S U) < < G (U) for every bounded U G with noncompact closure. There exists a large amount of literature devoted to measure of noncompactness and condensing operators (see, for example, 1,2,4, 6{8]). Let be a domain in R n. Let E bearegular space of-measurable functions on a domain here regularity means that every element i n E has an absolutely continuous norm. Let P D denote the operator of multiplication by the characteristic function D of a measurable subset D , i.e. P D u = D u. For bounded U E put (U) = E (U) = lim (D)!0 sup u2U kP D uk E for U E.