If $f\in L^\infty(X,\mu)$ which is the necesary and sufficient condition so that it is satisfied $\frac{1}{f}\in L^\infty(X,\mu)$?

Question

Let $f\in L^\infty(X,\mu)$. which is the necesary and sufficient condition so that it is satisfied $\displaystyle\frac{1}{f}\in L^\infty(X,\mu)$?

Answer 1

Let $f\in L^\infty(X,\mu)$. We prove: $$ \exists_{K>0}: |f(x)|>K \text{ almost everywhere } \Longleftrightarrow \frac{1}{f} \in L^{\infty}(X,\mu)$$ First we do $\Rightarrow$:

We have $0<K<|f(x)|$ for almost every $x$ hence $\frac{1}{|f(x)|}<\frac{1}{K}$. Hence $\frac{1}{f}\in L^{\infty}(X,\mu)$.

Now we do $\Leftarrow$:

There exists $M>0$ such that $\frac{1}{|f(x)|}<M$ for almost every $x$. Hence $|f(x)|>\frac{1}{M}>0$ almost everywhere. Take $K=M^{-1}$ and you are done.

Is this what you wanted?