From page 115 of "Measure theory and probability theory" by Krishna and Soumendra, as a step of the Radon-Nikodym theorem's proof:
Suppose that $\mu_1$ and $\mu_2$ are finite measures. Let $\mu$ be
the measure $\mu = \mu_1 + \mu_2$ and let $H = L^2(\mu)$. Define a linear function T on
H by
\begin{gather*}
T(f)=\int f d\mu_1
\end{gather*}
I don't get how $T$ can be well-defined on $L^2(\mu)$. Infact we have that $f \in L^2(\mu)$ implies:
\begin{gather*}
\int f^2 d\mu = \int f^2 d\mu_1 + \int f^2 d\mu_2 < \infty
\end{gather*}
That means $f \in L^2(\mu_1 )$ but not necessarely that $f$ is integrable. Thanks in advance.
Best Answer
$\int | f|d\mu_1 \leq \sqrt {\int f^{2} d\mu_1} \sqrt {\mu_1(\Omega)}$ by Holder's inequality.