Prelim problem in real analysis (any hints)

Question

Find all constants $K > 0$ for which the following holds:

If $(X,\Sigma,\mu)$ is any positive measure space and if $f:X\to \mathbb{R} $ is $\mu$ integrable satisfying $\left|\int_E f\,d\mu\right|<K$ for all $E\in \Sigma$, then $\|f\|_1<1$.

Any hints to start with?

Answer 1

Hint: Consider $E = \{x: f(x) \ge 0\}$ and $\{x: f(x) \le 0\}$ .