I'm following the book *Measure and Integral* of Richard L. Wheeden and Antoni Zygmund. This is problem 4 of chapter 8.

Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the integrals are taken over $E$, $1/p + 1/q=1$, with $1\lt p\lt \infty$.

I'm trying to prove that $$\int \vert fg\vert =\Vert f \Vert_p\Vert g \Vert_q$$ if and only if $\vert f \vert^p$ is multiple of $\vert g \vert^q$ almost everywhere.

To do this, I want to consider the following cases: if $\Vert f \Vert_p=0$ or $\Vert g \Vert_q=0$, we are done. Then suppose that $\Vert f \Vert_p\ne 0$ and $\Vert g \Vert_q\ne 0$. If $\Vert f \Vert_p=\infty$ or $\Vert g \Vert_q=\infty$, we are done (I hope). If $0\lt\Vert f \Vert_p\lt\infty$ and $0\lt\Vert g \Vert_q\lt\infty$, proceed as follows.

When we are proving the Hölder's inequality, we use that for $a,b\geq 0$ $$ab\leq \frac{a^p}{p}+\frac{b^q}{q},$$ where the equality holds if and only if $b=a^{p/q}$. Explicitly $$\int\vert fg \vert\leq \Vert f \Vert_p \Vert g \Vert_q \int\left( \frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}\right)=\Vert f \Vert_p \Vert g \Vert_q.$$ From here, we see that the equality in Hölder's inequalty holds iff $$\frac{\vert fg \vert}{\Vert f \Vert_p \Vert g \Vert_q}=\frac{\vert f \vert^p}{p\Vert f \Vert_p^p} + \frac{\vert g \vert^q}{q\Vert g \Vert_q^q}, \text{ a.e.}$$ iff $$\frac{\vert g \vert}{\Vert g \Vert_q}=\left( \frac{\vert f \vert}{\Vert f \Vert_p} \right)^{p/q},\text{ a.e.}$$ iff $$\vert g \vert^q\cdot \Vert f \Vert_p^p=\vert f \vert^p \cdot \Vert g \Vert_q^q,\text{ a.e.}$$ Q.E.D. But, assuming that $\Vert f \Vert_p\ne 0$ and $\Vert g \Vert_q\ne 0$, what about when $\Vert f \Vert_p=\infty$ or $\Vert g \Vert_q=\infty$? How can I deal with it?

In the case of Minkowski inequality, suppose that the equality holds and that $g\not \equiv 0$ (and then $\left( \int \vert f+g \vert^p\right)\ne 0$). I need to prove that $\Vert f \Vert_p$ is multiple of $\Vert g \Vert_q$ almost everywhere. I can reduce to the "Hölder's equality case". I can get $$\vert f \vert^p=\left( \int \vert f+g \vert^p\right)^{-1}\Vert f \Vert_p^p\vert f+g \vert^p$$ $$\vert g \vert^p=\left( \int \vert f+g \vert^p\right)^{-1}\Vert g \Vert_p^p\vert f+g \vert^p$$ almost everywhere, but again, using the finiteness and nonzeroness of $\Vert f \Vert_p$ and $\Vert g \Vert_p$.