M. Pal [Glas. Mat., III. Ser. 20(40), 327-335 (1985; Zbl 0615.28001)] proved several results on Lebesgue measurable sets \(A\subset {\mathbb{R}}^ n\) having Lebesgue measure \(m(A)>0.\) The author obtains extensions of some of Pal’s results. One of these reads as follows.
Let \(A\subset {\mathbb{R}}^ n\) be a set of positive Lebesgue measure. Let \((b_ k)\subset {\mathbb{R}}\) be any sequence such that \(b_ k\neq 0,\) \(k\geq 1,\) and the limit \(\lim b_ k\) exists and is non-zero. Let W be a metric space, \(w_ 0\in W,\) and let \(T_ w: {\mathbb{R}}^ n\to {\mathbb{R}}^ n,\quad w\in W,\) be transformations such that each \(T_ w\) maps measurable subsets of \({\mathbb{R}}^ n\) into measurable subsets of \({\mathbb{R}}^ n\). Assume further that \[ m(T_{w_ 0}(B))>0\quad whenever\quad m(B)>0\text{ and } \lim_{n\to \infty}m(T_{w_ 0}(A\cap K)\setminus T_{w_ n}(A\cap K))=0 \] for each ball K and each sequence \((w_ n)\subset W\) converging to \(w_ 0\). Let \((w_ n)\subset W\) converge to \(w_ 0\). Then, for each natural number p, there exists a ball \(K_ p\) centered at the origin, a positive integer \(N_ p\) such that for each sequence of p integers \(N_ p<n_ 1<...<n_ p\) there exists a sequence \((n_ i),\quad i\geq 1,\) and a subsequence \((b_{k_ i})\) of \((b_ k)\) satisfying \(b_{k_ i}=b_ i,\quad 1\leq i\leq p,\) with the following property: For each \(x\in K_ p\) there exist points \(a_ 1,a_ 2,...\) in A such that \[ x=(T_{w_{n_ i}}(a_ i)-T_{w_ 0}(a_ 0))/b_{k_ i},\quad i=1,2,.... \] The author also obtains Baire set analogues of these extensions.