I'm reading this proof where they are trying to show that the Holders inequality becomes equal iff $|f|^p$ and $|g|^q$ are multiples of each other.
I'm stuck in this step:
$$\int\vert fg \vert\leq \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)=\Vert f \Vert_p \Vert g \Vert_q.$$ From here, we see that the equality in Hölder's inequalty holds iff $$\frac{\vert fg \vert}{\Vert f \Vert_p \Vert g \Vert_q}=\frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}, \text{ a.e.}$$
I'm not sure how they got the forward direction. Suppose Holders' inequality is equal, so
$$\int\vert fg \vert = \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)$$
Rearranging gives me
$$\int \frac{\vert fg\vert}{\Vert f \Vert_p \Vert g \Vert_q }= \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)$$
This only tells me that the integrals are equal, but how does that tell me the integrands are equal almost everywhere? I don't see how the equation above tells me $\frac{\vert fg \vert}{\Vert f \Vert_p \Vert g \Vert_q}=\frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}, \text{ a.e.}$
This is actually explained in the comments. If the integral of a nonnegative function is zero, then the function must be zero. To see why it's nonnegative, by Young's Inequality,
$$ \frac{\vert fg\vert}{\Vert f \Vert_p \Vert g \Vert_q } = \bigg|\frac{|f|}{\Vert f \Vert}_p \frac{|g|}{\Vert g \Vert}_q \bigg| \leq \frac{1}{p} \bigg(\frac{|f|}{\Vert f \Vert}_p \bigg)^p + \frac{1}{q} \bigg(\frac{|h|}{\Vert h \Vert}_q \bigg)^q = \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q} $$.
Therefore, the function must be zero.