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If I correctly understand, you are misinterpreting the meaning of the product and sum of observables.

When you say "We can now define a sum and a product of observables. These are obtained by performing the two measures and then adding or multiplying their values."

This cannot possibly describe the usual sum A+B and product AB of operators. For the product, it is not even hermitian unless A and B commute. Agreed, A+B is hermitian, but the spectrum of A+B does not contain the result of the sum of a measurement of A followed by a measurement of B (in either way), again unless A and B commute. For a counter-example take A=[[1 0];[0 -1]] $A=\pmatrix{1& 0\cr 0&-1}$ and B=[[0 1];[1 0]].$B=\pmatrix{0&1\cr 1&0}$.

I hope I correctly understood your question.

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If I correctly understand, you are misinterpreting the meaning of the product and sum of observables.

When you say "We can now define a sum and a product of observables. These are obtained by performing the two measures and then adding or multiplying their values."

This cannot possibly describe the usual sum A+B and product AB of operators. For the product, it is not even hermitian unless A and B commute. Agreed, A+B is hermitian, but the spectrum of A+B does not contain the result of the sum of a measurement of A followed by a measurement of B (in either way), again unless A and B commute. For a counter-example take A=[[1 0];[0 -1]] and B=[[0 1];[1 0]].

I hope I correctly understood your question.