1) Is every mesurable function in $L^P(\mathbb{R}^N)$
2) is the standard mollifiers in $L^P(\mathbb{R}^N)$
here $L^P(\mathbb{R}^N)=\{ f|$ is measurable and $\int |f|^p <\infty \}$
and here molllifiers with property that $\int \rho _\epsilon (x) dx =1$
here i think $1/x$ is measurable but not $L^1$ am i correct for question (1)
This is for part 1)
Not necessarily. Let $E$ be any Borel measurable set with infinite measure and consider the characteristic function of this set. That is, consider $f(x)=\chi_E$.
Then $f$ is measureable (exercise)
however $f$ is not in $L^p$ for any $p \geq 1$
since $\int |f|^p = \int (\chi_E)^p = \int \chi_E = m(E) = \infty$