If a multiple integral is zero over some region, can I say the integrand is zero? Consider the following problem
Let the integral of a real function of 3 real variables $F(x,y,z)$ over some volume $V$ of $R^{3}$ vanish, 
$\int$$\int$$\int$$dxdydz$ $F(x,y,z)$$=0$ 
Now assume this continues to hold for any finite volume $V$ contained in a larger volume $V'$ of $R^{3}$. 
Can I say the integrand is necessarily zero over $V'$? 
This seems to be valid for definite integrals on a line (although I'm also not sure), and I feel the function would have to vanish, but I don't see how to prove this. 
 A: Sure, the function must be zero almost everywhere, under the minimal assumption that it is measurable. Note that these are very reasonable and non-restrictive conditions:


*

*The almost everywhere condition is required since you can modify a function on a measure zero (i.e., zero volume) set to be anything you like, without changing any integrals.

*The measurability condition is required for the integrals to even be well-defined.


Although the question is stated in $\mathbb{R}^3$, the result actually holds on measure spaces more generally, and in my opinion the reasoning behind the result is easier to understand in this more general setting.
The argument is straightforward. Suppose for the sake of contradiction that the integral of your function over any measurable set is zero, yet the function itself is strictly nonzero on some set $V$ with positive measure. 
Decompose $V$ into the union of a piece where the function is strictly positive and another piece where the function is strictly negative. Since our function is measurable, these pieces are measurable (they are the intersection of a measurable set with the pullback of a measurable set under a measurable function). Furthermore, at least one of these pieces must have positive measure, since their union is $V$, which has positive measure. The integral over that piece must then be nonzero*, a contradiction.

*For a proof that the integral of a strictly positive (or strictly negative) function over a set of positive measure is nonzero, see the answer to the question here: Lebesgue integral of a positive function on a set of positive measure . The basic premise is to decompose your set into a countable union of subsets where your function is bounded away from zero by successively smaller and smaller tolerances. Since the countable union of measure zero sets has measure zero, at least one of these subsets must have positive measure, and so the integral on that subset is strictly nonzero.

For some further intuition about this result and why it should be true physically, imagine that your function represents temperature in a room. If you measure the temperature in the room with a thermometer, you don't get the exact temperature at a point - rather, you get a local average temperature over a small region of space around the thermometer tip. This local "average temperature" is essentially the integral of your function over that set of space, divided by its volume. The more spatially accurate the thermometer, the smaller the volume of integration.
If the integral of the temperature function is zero over all volumes of space, that is like saying that any thermometer you use to measure it, no matter how accurate, will always say that the room has temperature zero. There is no physical measurement possible that can distinguish the temperature field from the zero field.
