The following question is in regard to a line from the textbook `Interpolation Spaces' by Bergh and Lofstrom. More precisely, it is in regard to the last line on page 110, in the proof of Theorem 5.2.2.
Let $(\Omega,\mathcal{A},\mu)$ be a measure space and $0<p_{0}<p_{1}<\infty$, and suppose that $f\in L^{p_{0}}(\Omega,\mathcal{A},\mu)+L^{p_{1}}(\Omega,\mathcal{A},\mu)$ (these can be either real or complex $L^{p}$ spaces). Bergh and Lofstrom then claim that for any $t>0$ one has that
$$\inf_{\substack{f=f_{0}+f_{1} \\ f_{0}\in L^{p_{0}} \\ f_{1}\in L^{p_{1}} }}\int_{\Omega}|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}d\mu(x)=\int_{\Omega}\inf_{\substack{f(x)=f_{0}(x)+f_{1}(x) \\ f_{0}\in L^{p_{0}} \\ f_{1}\in L^{p_{1}} }}\big(|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}\big)d\mu(x).$$
So, my question is how does one prove this? Of course, if $f_{0}\in L^{p_{0}}$ and $f_{1}\in L^{p_{1}}$ with $f=f_{0}+f_{1}$ then clearly
$$|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}\geq\inf_{\substack{f(x)=f_{0}(x)+f_{1}(x) \\ f_{0}\in L^{p_{0}} \\ f_{1}\in L^{p_{1}} }}\big(|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}\big)$$
for all $x\in\Omega$ and so
$$\inf_{\substack{f=f_{0}+f_{1} \\ f_{0}\in L^{p_{0}} \\ f_{1}\in L^{p_{1}} }}\int_{\Omega}|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}d\mu(x)\geq\int_{\Omega}\inf_{\substack{f(x)=f_{0}(x)+f_{1}(x) \\ f_{0}\in L^{p_{0}} \\ f_{1}\in L^{p_{1}} }}\big(|f_{0}(x)|^{p_{0}}+t|f_{1}(x)|^{p_{1}}\big)d\mu(x).$$
However, I am stuck trying to prove the reverse inequality. Any help or any suggestions on how to do this would be greatly appreciated.