Implication of a set of zeros with positive measure

[noowutah]

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

 

[mfb]

A non-zero ψ_i for at least one i would allow to solve the sum equation for ...
This will eventually lead to a contradiction with "non-zero measure".