How to understand if a function is in $L^p$ space?

Question

I'm studying measure theory and I'm having some trouble understanding which functions are in $L^p$ space. For example the constant function $f(x) = c$ is in $L^1$ space? Following the definition I would say (in general) no because its integral over $\mathbb{R}$ doesn't converge. I would say the same for $x^2$. Am I wrong? Does something change if I'm working with a finite (positive) measure like the probability measure?

Answer 1

To talk about a $L^p$ space, one would first need to define a measure space $X$ and a measure $\mu$. Therefore, the notation $L^p(X,\mu)$ is common. Saying a function $f$ is $L^p$ is meaningless until the measure space and measure is specified. Even for a function $f:\mathbb{R}\to \mathbb{R}$, it might be $L^p$ with respect to one measure but not with respect to another.

The definition is $$L^p(X,\mu) = \lbrace f:X\to\mathbb{R} \text{ measurable}\mid \int_X|f(x)|^pd\mu(x)<\infty\rbrace $$ Considering your examples, it is now clear that $f_c:\mathbb{R}\to \mathbb{R}$ given by $f_c(x) = c$, a constant, is not $L^1$ with respect to the Lebesgue measure, as the integral will be infinite (EDIT: unless $c=0$, the zero function is always $L^p$ for all $p$). The same holds for any polynomial. But for a probability measure, $f_c\in L^1$, as the integral will be $c$.