$f\in L^p(X,\mu)$ , $f-1\in L^q(X,\mu)$ then $\mu(X) < \infty $

Question

Can some one give a hint how to start to solve :

Assume $ 1 \le p,q < \infty $ and $$f\in L^p(X,\mu)$$ now if we assume $$f-1\in L^q(X,\mu)$$ then we have $$\mu (X) < \infty $$

Thanks

If $|f|$ or $|f-1|$ has a positive infimum $m$ then problem is solved because:

$ m^p \mu(X)^p < (\int_{X} |f|^p \,\ dx)< \infty $

Answer 1

Without loss of generality (exchanging $f$ and $f-1$) we can assume $q\geq p$.

We are given that $$\infty>\int|f|^p=\int_{\{|f|^p<1\}}|f|^p+\int_{\{|f|^p\geq 1\}}|f|^p$$

Since $M\mu(\{|f|^p\geq M\})\leq\left|\int_{\{|f|^p\geq M\}}|f|^p\right|$, in particular $$\mu(\{|f|^p\geq 1\})<\infty$$

Now $$\infty>\int_{\{|f|^p<1\}}|f|^p\geq\int_{\{|f|^p<1\}}|f|^q$$

By Minkowsky's inequality

$$\begin{align}\mu(\{|f|^p<1\})^{1/q}&=\left(\int_{\{|f|^p<1\}}1\right)^{1/q}\\&=\left(\int_{\{|f|^p<1\}}|f+1-f|\right)^{1/q}\\&\leq\left(\int_{\{|f|^p<1\}}|f|^q\right)^{1/q}+\left(\int_{\{|f|^p<1\}}|1-f|^q\right)^{1/q}\\&<\infty\end{align}$$

Therefore

$$\mu(X)=\mu(\{|f|^p<1\})+\mu(\{|f|^p\geq 1\})<\infty.$$