Measurable selection theory deals with the following problem: Let $(X,\germ A)$ be a measurable space, $Y$ a topological space and $F$ a map from $X$ to the nonempty subsets of $Y$. Under what conditions is there a measurable selection for $F$, i.e., an $\germ A$-measurable map $f$ from $X$ to $Y$ such that $f(x)\in F(x)$ for all $x$? Usually the conditions require some kind of measurability of $F$, often completeness properties of the values $F(x)$ and---roughly---metrizability properties of $Y$. In ``classical'' selection theory a further crucial condition is that $Y$ be separable. (For a survey see the articles by D. H. Wagner [SIAM J. Control Optim. {\bf 15} (1977), 859--903; MR0486391 (58 \#6137); {\it Measure theory\/} (Oberwolfach, 1979), pp. 176--219, Lecture Notes in Math., 794, Springer, Berlin, 1980; MR0577971 (83c:28009)].) Recently, successful attempts have been made to remove this condition when more is known about $(X,\germ A)$ and $F$. (An---incomplete---list of references is given at the end of this review.) The paper under review belongs to this line of research. Here the basic assumption is that $\germ A$ is the $\sigma$-algebra of $\mu$-measurable sets, where $\mu$ is a perfect measure, and that $F(X)=\bigcup\{F(x)\colon x\in X\}$ has cardinality smaller than the first measurable cardinal. Theorems 3.1 and 4.1 of the paper can be combined into the following theorem: Let $\mu$ be a perfect measure on a set $X,\germ A$ be the $\sigma$-algebra of $\mu$-measurable subsets of $X$, and $Y$ be a metric space. Let the set-valued map $F$ satisfy the cardinality condition above. Then $F$ admits an $\germ A$-measurable selection in each of the following two cases: (i) $X$ is a metric space, $\mu$ is a Borel measure, $F$ is lower semicontinuous, i.e., $\{x\in X\colon F(x)\cap U\neq\varnothing\}$ is open for $U\subseteq Y$ open, and $F(x)$ is complete for all $x$. (ii) $F$ is weakly measurable, i.e., $\{x\in X\colon F(x)\cap U\neq\varnothing\}\in\germ A$ for $U\subseteq Y$ open, and $F(x)$ is compact for all $x$. Further results of the paper concern selections measurable with respect to other $\sigma$-algebras. \par \{Reviewer's remarks: Corollary 5.1 remarks that the theorem applies in particular to locally compact spaces $X$ with a Radon measure $\mu$. The result following from case (ii) has a much stronger version due to D. H. Fremlin [{\it Measure theory\/} (Oberwolfach, 1981), Problem section, pp. 425--428, Lecture Notes in Math., 945, Springer, Berlin, 1982], who showed that (ii) implies the existence of a measurable selection when $\mu$ is a Radon measure on an arbitrary Hausdorff space without any cardinality condition on $F(X)$. A result related to case (i) was recently proved by J. E. Jayne and C. A. Rogers [Acta Math. {\bf 149} (1982), 87--125]. They showed that every upper semicontinuous map from a metric space to the nonempty subsets of another metric space has a selection of the second Borel class. See also papers by Z. Frolík and P. Holický [Comment. Math. Univ. Carolin. {\bf 21} (1980), 653--661; MR0597756 (83c:54026)], R. W. Hansell [``Hereditarily-additive families in descriptive set theory and Borel measurable multimaps'', Preprint, 1981; per revr.] and J. Kaniewski and R. Pol [Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. {\bf 23} (1975), 1043--1050; MR0410657 (53 \#14405)]. A kind of survey article by S. Graf has appeared [Rend. Circ. Mat. Palermo (2) {\bf 1982}, Suppl. No. 2, 87--122].\}