let $(\Omega ,A,\mu)$ be a measure space and let $1<p<q<r<\infty$. show that $$L^q(\Omega)\subset L^p(\Omega)+L^r(\Omega)$$
answer:Iknow that I can assume that $f\in L^q(\Omega)$,$E=\{\omega \in\Omega : \mid f(\omega )\mid >1\}$ and decompose fby $f1_E+f1_(\Omega$\E) but I donot know how?
Suggestion: $E$ is a set where $f$ is "large". When $f > 1$, $|f|^p$ increases and decreases in direct relation to $p$. So if you know $f$ integrates at the $q$th power, what other types of powers will be integrable on this set?
Likewise when $|f| < 1$ we are talking about the "tail" of the integral if $\Omega$ has infinite measure, or simply the region where $f$ is small on a finite $\Omega$. Here, increasing the exponent increases integrability. For a hopefully helpful mental example, consider $\int_1^{\infty} \frac{1}{x} \, dx$ vs $\int_1^{\infty} \frac{1}{x^2} \, dx$.
Hopefully you can use this idea to show that $f 1_E \in L^p$ and $f 1_{E^c} \in L^r$. But $f = f 1_E + f 1_{E^c}$, so $f$ is a sum of a $L^p$ function and a $L^r$ function. Since $f$ was a general member of $L^q$, do you see how we have shown the desired inclusion?