Integrands that vanish everywhere

[StatusX]

In physics, we often use the assumption that if the integral:

$$\int_D f(\vec x) d \vec x =0$$

and this holds for any region D in the space, then the integrand must vanish identically everywhere in the space. This was motivated by the argument that if it didn't vanish in some region, we could focus the integrand in that region (say, only where it is positive) and the above equation wouldn't hold. But I was thinking, wouldn't this argument only show that the integrand vanishes almost everywhere. That is, it could still be non-zero on a set of measure zero, and the above equation would still always be satisfied. Is it just that this possibility doesn't concern physicists, or is there actually a way to show it must vanish at every point?

 

[StatusX]

Actually, now I see that the argument would work if the function were continuous, because then if there was any point x where f(x) were non-zero, say positive, it would have a neighborhood where f was positive, and so we could integrate over that neighborhood to get a non-zero value. That's good enough for me. You can delete this thread if you want.

 

[tacman]

This is an interesting question and brings up an important point in mathematical reasoning and the use of assumptions in physics. It is true that in mathematics, the assumption that an integral vanishes for all regions would only imply that the integrand vanishes almost everywhere. However, in physics, we often make certain assumptions and simplifications in order to make our calculations and models more manageable. In this case, the assumption that the integrand vanishes everywhere is a simplification that is often used in physics because it simplifies the analysis and allows us to make predictions and draw conclusions more easily.

In addition, in physics, we are often dealing with continuous functions and physical quantities that have a physical meaning and interpretation. Therefore, even if the integrand is non-zero on a set of measure zero, it would not have any physical significance and would not affect our conclusions and predictions. This is because physical measurements and observations are always subject to some level of uncertainty and error, and we can only measure and observe physical quantities up to a certain precision.

Furthermore, in physics, we often use a combination of mathematical reasoning and experimental data to validate our assumptions and models. If the assumption that the integrand vanishes everywhere leads to predictions and results that are consistent with experimental data, then it is considered a valid assumption in the context of physics.

In summary, while the mathematical argument may suggest that the integrand only vanishes almost everywhere, the assumption that it vanishes everywhere is a simplification that is often used in physics for practical and pragmatic reasons. As long as this assumption leads to consistent and accurate predictions, it is considered a valid assumption in the context of physics.