I stumbled onto this exercise while studying for an exam. I thought it looked fun.
Let $f$ be integrable over $\mathbb{R}$. Show the following four assertions are equivalent.
(i) $f=0$ a.e. on $\mathbb{R}.$
(ii) $\int_\mathbb{R} fg=0$ for every bounded measurable function $g$ on $\mathbb{R}.$
(iii) $\int_A f =0$ for every measurable set $A$.
(iv) $\int_O f = 0$ for every open set $O$.
Just a few observations.
(i) implies (ii) Does this follow from how the function is defined and this inequality: $\int_\mathbb{R} fg \leq \int_\mathbb{R} f\cdot M =0$.
(iii) implies (iv) does that work out because for $\int_E f = 0$ iff $f=0$ a.e. on $E$ and the fact that open sets are also measurable sets.
I don't actually want a rigorous proof of this. I'm just interested in fine tuning my intuition about integrable functions and pulling together definitions.
Edit: I do not have a strong intution of going from (ii) to (iii).