equivalent definitions of integrability

Question

Is the following equivalent:

1) A function $f\colon\mathbb{R}^n\to (-\infty,\infty]$ is Lebesgue integrable if there exists a sequence of step functions $(g_n)_n$ such that $\|f-g_n\|_1\to\infty$.

2) $\|f\|_1<\infty$ ?

From 1 to 2 I can explain it to myfelf using that $(\int |g_n| dx)_n$ is a cauchy secuence which converges to $\int |f| dx$. However, how does 2=>1work?

Answer 1

Use the standard construction of simple functions converging to $f$: $g_n(x)=\frac {i-1} n$ if $ \frac {i-1} n \leq f(x) < \frac i n$ and $|i| \leq n^{2}$, $0$ if $|i| > n^{2}$. Then $\int |g_n-f| \to 0$ and each $g_n$ is simple.