A transversal, or marriage, is an injective choice function from a set A to a set B with: graph contained in a given set S. P. Hall [1] gave necessary and sufficient conditions on S for the existence of a marriage when these sets are finite. His theorem has been extended to infinite sets where the of a set is its cardinality; see [4] and [5]. This kind of function arises in many branches of analysis, but usually the correct notion of the size of a infinite set is its Lebesgue measure. Also, instead of being injective, the functions are required to be measurepreserving. Measure-preserving transformations of the unit interval are important in ergodic theory; see Halmos [2]. In real analysis, they often arise as a change of variables; see Royden I-6]. In harmonic analysis and differential equations, they are often called rearrangements; see Stein [7]. It is natural to ask about the existence of such transformations. Examples given below show that any analog of Hall's theorem involving Lebesgue measure must include a certain incompleteness, or error. This paper provides two such analogs. In each case, it is shown that the error can be made arbitrarily small. So, for most analytical purposes, the error is negligible. The author has shown in [-3] that the hypotheses of the marriage theorem are closely related to the important class of weak-type linear operators [7]. To make this more precise, let I [ 0 , 1] equipped with the usual Lebesgue measure. Suppose S c I x I (which represents all (x, y) such that x is compatible with y). If J~_I, let E ( J ) = {yEI: ( x , y )~S , for some x e J}. We will call ~b a marriage on I within S if it is a bijection from I to I such that for all x~I , (x, ~ ( x ) ) e S and for all J~_I, Iff(J)J = IJJ. The natural conjecture would be the following;