$L(x) = 1 - F(x)$ is the slowly varying upper tail, where F(x) is a continuous probability distribution function with infinite expectation. That is $\int_{0}^{\infty} u dF(u) = \infty$. I am unable to construct any such probability distribution and would like some examples with the specified properties.
Definition: A function $L(x)$ is said to be slowly varying at $\infty$ if $lim_{x \to \infty} \frac{L(\alpha x)}{L(x)} = 1 , \forall \alpha >0$, where $\alpha$ is a real number.
Consider the cdf $F(x) = 1-\frac{1}{\log(x)}$ for $x\geq e$. Then $L(x) = \frac{1}{\log(x)}$ is slowly varying at infinity and if $X$ is a random variable with cdf $F$ then $$\mathbb{E}[X] = \int_0^\infty \mathbb{P}(X>x) \, dx = \int_0^e 1 \, dx + \int_e^\infty \frac{dx}{\log(x)} = \infty.$$