Let $(X,\mathcal{F},\mu)$ a measurable space. Let $f,g: X\rightarrow\bar{\mathbb{R}}$ two measurable functions in X.
The sum of two measurable functions is well defined (and measurable) only where there is no $+\infty -\infty$ situation (which may occurs because in measure theory we're working on the real extended number). In this point the sum is not defined.
The fact is that I cannot understand how we can say that the spaces $\mathcal{L}^p(X)$ are vector space. I mean the sum of two elements in $\mathcal{L}^p(X)$ is a function that is generally measurable only in subset of X.
Hope is clear.
Thanks