We discuss boundedness and compactness properties of the embedding M Λ 1 ⊂ L 1 ( μ ) , where M Λ 1 is the closed linear span of the monomials x λ n in L 1 ( [ 0 , 1 ] ) and μ is a finite positive Borel measure on the interval [ 0 , 1 ] . In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences Λ . Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from M Λ 1 to L 1 ( [ 0 , 1 ] ) .