Suppose I am given a function $f$ which is integrable. Is it possible to formulate the following Fundamental Lemma for Variational Calculus?
If for all measurable integrable function $\phi$ and a integrable function $f$:$$\int_{-\infty}^{\infty}f(x)\phi(x)dx=0$$, then $f=0$ a.e..
My suggestion to proof this is, if $f\neq0$, then there exist a measurable set $A$ such that $f>0$. Taking the characteristic function $1_{A}$, which is clearly measurable, then $$\int_Af(x)dx>0$$ which is a contradiction.
Is this correct? Do I need the boundedness of the set $A$?