For $E \subset \mathbb{R^n}$, is $f^p$ finite almost everywhere in $E$ if $f \in L^p(E)$?

Question

Q1)

I didn't learn $L^p(E)$ yet, only learned $L(E)$. In order to solve problems, however, I need to know that the following theorem is correct or not.

Let $E \subset \mathbb{R^n}$.

Then, if $f \in L^p(E)$, $f^p$ is finite almost everywhere in $E$.

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Q2)

My textbook said \begin{align}f\in L(E)&\Longleftrightarrow\int_E f \quad\text{is finite}\\ f\in L^{p}(E) &\Longleftrightarrow \int_E |f|^p\lt\infty\end{align}

Why isn't absolute used in $L(E)$ while absolute is used in $L^p(E)$?

Answer 1

$$L^p(E)=\{f: \int_E \vert f \vert^p <\infty \} $$

So $ f\in L^p(E) \rightarrow f^p \in L^1(E)$

Then you can prove your question 1 by contraddiction! ($f^p = \infty$ on a positive measure set then...)

About question 2, I thin you're right, there must be an absolute value!