The subset of $L_p$ space has empty interior

Question

Let $f\in L_1([0,1], \lambda)$ and suppose $\int_0^1 |f|^2 dx \leq M$.

Does there exist function $g \in L_1([0,1], \lambda)$ which is close to $f$ (i.e. $\|f-g\|_{L_1} < \epsilon$) but which is not square-integrable, or at least satisfies $\int_0^1 |g|^2 dx > M$?

Answer 1

Take your favorite function $h \in L^1(0,1) \setminus L^2(0,1)$. Then set $g_t:=f+th$, for $t\searrow 0$, $\|f-g\|_{L^1} \searrow0$, but $g_t\not\in L^2(0,1)$.

(One valid choice of $h$ would be $h(x) =x^{-1/2}$.)