Iff is a bounded Lebesgue measurable function on [0, 1] and 1 < p < 00, let fp denote the best Lp-approximation to f by nondecreasing functions. It is shown that fp converges almost everywhere as p decreases to one to a best Ll-approximation to f by nondecreasing functions. The set of best Ll-approximations to f by nondecreasing functions is shown to include its supremum and infimum. Let Q = [0, 1], yt = Lebesgue measure and W = the Lebesgue measurable subsets of Q2. For 1 < p < oo, let Lp = Lp(Q, X, ,t). Let M denote the set of all nondecreasing functions on Ci. Suppose f E L,. For 1 < p < so, Lp is a uniformly convex Banach space and M is a closed convex subset thereof, so f has a unique best Lp-approximation fp by elements of M, i.e., fp is the unique element of M which satisfies lIf-fpllp = inf{lfhllp: h E Ml. The function f is said to have the Polya property if limp -OOp exists almost everywhere as a bounded measurable function and the Polya-one property if limp 1 fp exists in the same way. The Polya property fails for an arbitraryf in L., [2, 1] but, as is shown in this note, the Polya-one property obtains. If B is a subset of L1, let tt1(f1B) denote the set of all best L,-approximations of f in B and let f(B) = inf t1(flB),f(B) = sup tt1(fIB). If e is a subsigma algebra of W and B is the subspace of L1 consisting of all u-measurable functions, then f(B) and f(B) are in tt1(flB) and g E t 1(flB) if and only if f(B) < g < f(B) [5]. Let f = f(M) and f= f(M). In this note we show thatf, f E t1(flM) and that every convex combination off and f is in tt1(fIM) (so that f and f are extreme points of the L,-compact convex set ttl(fI M)), but there may be a function g E M such that f< g < f but g is not in tt 1(fl M). LEMMA 1. M is an L1-closed convex subset of L1, and tt1(flM) is a nonempty subset of L,. PROOF. Suppose {gn: n = 1,2,...} c M and gn -_ g in L1. Since {gn} has a subsequence which converges to g almost everywhere, we may assume that gn g almost everywhere. Letg = lim supn gn. Since each gn is nondecreasing, g is nondecreasing. Thus g is equivalent to an element of M. Clearly M is convex. Received by the editors February 14, 1984. 1980 MVathematics Subject Classification. Primary 41A30, 41A50; Secondary 40A30, 26A48.