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## Main Question or Discussion Point

I have a non-zero measured subset [itex]X\subseteq\mathbb{R}^{n}[/itex] on which [itex]\sum_{i=1}^{n}\psi_{i}x_{i}=0[/itex] for all [itex]x=(x_{1},\ldots,x_{n})[/itex] in [itex]X[/itex]. How can I show that [itex]\psi_{i}=0[/itex] for [itex]i=1,\ldots,n[/itex]?