A non-zero measured subset X in R^n satisfies the equation sum(ψ_ix_i) = 0 for all x in X. To demonstrate that each ψ_i must equal zero, one must show that if any ψ_i were non-zero, it would contradict the non-zero measure of X. The existence of a non-zero ψ_i would imply the ability to solve the sum equation for specific x values, leading to a logical inconsistency. Therefore, all coefficients ψ_i must be zero to maintain the integrity of the non-zero measure condition. This conclusion reinforces the relationship between the properties of measurable sets and linear combinations.
