A multidimensional Lyapunov type theorem

Given functions ${\displaystyle f_1,...,f_{\nu }\in {ℒ}^1({ℝ}^n;{ℝ}^m)}$, weights ${\displaystyle p_1,...,p_{\nu }:{ℝ}^n\to [0,1]}$ with ${\displaystyle \sum p_i\equiv 1}$, and any finite set of vectors ${\displaystyle v_1,...,v_k\in {ℝ}^n\setminus \left\{0\right\}}$, we prove the existence of a partition ${\displaystyle \{A_1,...,A_{\nu }\}}$ of ${\displaystyle {ℝ}^n}$ such that the two functions ${\displaystyle f_p={\sum }_{i=1}^{\nu }{p}_i{f}_i,}$f_A = ∑_{i=1}^ν χ_{A_i}f_i${\displaystyle havethesame\int egral\neg onlyover}$ℝ^n${\displaystyle ,butalsooverevery\sin g\le l\in e}$x' + ℝv_j${\displaystyle ,f\text{or}eachj=1,...,k\text{and}almosteveryx\prime \in the\text{or}thogonalhyperpla\ne }$v_j^⊥${\displaystyle .Equiva\le ntly,theFouriertran\text{or}msof}$f_p${\displaystyle ,}$f_A${\displaystyle satiy}$f̂_p(y) = f̂_A(y)${\displaystyle f\text{or}every}$y ∈ ⋃ v_j^⊥!$!.