$L_p$ function and improper integral

I was studying at $L_p$ spaces, and I came across this thread (among others) : Show that the following function $f \in L_p$ if and only if $p=2$

Now, my question is : Why they analyze if the function is $L_p$ using Riemann Improper integration, when the user is asking in terms of Lebesgue Measure? The fact that a Riemann improper integral is not finite implies that the Lebesgue integral is not finite?

Sorry if the question is too decontextualized or direct but I don’t understand that.

Thanks.