Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \langle f(t), v(t) \rangle_{V',V}$$ holds for almost every $t \in [0,T].$ Here $A:V \to V'$ is some operator (eg. Laplacian).

Presumably the null set on which this equation doesn't hold is the same for every $v.$ What would happen in the null set is different for every $v$? How would the theory change?

I have never seen null sets being discussed, except for one line in Zeidler where he says the null sets are the same. A similar question was asked here: http://math.stackexchange.com/questions/418693/need-explanation-of-passage-about-lebesgue-bochner-space

Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \langle f(t), v(t) \rangle_{V',V}$$ holds for almost every $t \in [0,T].$ Here $A:V \to V'$ is some operator (eg. Laplacian).

Presumably the null set on which this equation doesn't hold is the same for every $v.$ What would happen in the null set is different for every $v$? How would the theory change?

I have never seen null sets being discussed, except for one line in Zeidler where he says the null sets are the same. A similar question was asked here: https://math.stackexchange.com/questions/418693/need-explanation-of-passage-about-lebesgue-bochner-space

Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u', v \rangle + b(u,v) = \langle f, v \rangle$$ holds for almost every $t \in [0,T].$

Presumably the null set on which this equation doesn't hold is the same for every $v.$ What would happen in the null set is different for every $v$? How would the theory change?

I have never seen null sets being discussed, except for one line in Zeidler where he says the null sets are the same. A similar question was asked here: http://math.stackexchange.com/questions/418693/need-explanation-of-passage-about-lebesgue-bochner-space

Consider the weak formulation: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), v(t) \rangle_{V',V} + \langle Au(t), v(t) \rangle_{V',V} = \langle f(t), v(t) \rangle_{V',V}$$ holds for almost every $t \in [0,T].$ Here $A:V \to V'$ is some operator (eg. Laplacian).

Presumably the null set on which this equation doesn't hold is the same for every $v.$ What would happen in the null set is different for every $v$? How would the theory change?

I have never seen null sets being discussed, except for one line in Zeidler where he says the null sets are the same. A similar question was asked here: http://math.stackexchange.com/questions/418693/need-explanation-of-passage-about-lebesgue-bochner-space