Integrability of a function in $L^2$ we consider $V $ a polynomial in $R[x_1,x_2,..,x_n]$ such that $e^{-V(X)}\in L^2(R^n)$
I want to prove that this implies that $e^{-V(X)} $ vanish as $|X|$ goes to $+\infty$
For that we suppose that  the function didn't vanish,and we prove  that this implies that 
it's value would be greater than ε>0 in a set of an infinite measure.
But i didn't know how to prove that .Can someone help me please

If you are taking the Lebesgue integralof $f$ then what $f$ does on a  set  of measure $0$ is irrelevant.  So if $S$ is an unbounded  null set, let $f(x)=0$ for $x\not \in S,$ and let $f(x)$ be anything  at all for $x\in S.$ Then $\int f=0.$ Example: Let $S=Q^n$ and let $f(x_1,...,x_n)=\sum_{j=1}^n|x_j|$ for $(x_1,...,x_n)\in S$ and $f(x)=0$ for $x\not \in S.$ And $\int_{R^n}f=0.$
