Existence of sequence of regular projections

Question

Reading the book :Krasnosel'skii, M.A.; Pustylnik, E.I.; Sobolevskii, P.E.; Zabreiko, P.P. (1976), Integral Operators in Spaces of Summable Functions, Leyden: Noordhoff International Publishing, 520 p., section 1.5, i found the definition of regular projection. Let some $1\leq p\leq \infty$, and consider the space $L_p(\Omega,\mu)$ for some $\Omega$. Suppose that $$\Omega=\Omega_1\cup\cdots\cup \Omega_r,$$ where the union are disjoint. Then we define the operator $$Pf(t)=\mu(\Omega_i)^{-1}\int_{\Omega_i}f(s)ds,$$ if $t\in \Omega_i$, $i=1,\ldots,r$ and $f\in L_p$. Clearly $P$ is well defined, and has finite rank. If now we consider a sequence of this type of partitions $$\Omega=\Omega_i^n\cup\cdots\cup \Omega_{q(n)}^n,$$ for any $n$ natural, where $q$ is some function of $n$, then we say that the associated sequence of operators $\{P_n\}$ is regular if the maximum diameter of the set $\Omega_i^n$ tends to $0$ as $n\to \infty$. Then, the important result is that $P_n\to I$, i.e., the sequence converges strongly to the identity. This all is used in the proof of Riesz-Thorin theorem with compactness, but in the proof he assumes that such a sequence of regular projections exists, and doesn't elaborate further. Why is that? It does not depend if we can split the domain $\Omega$ in a disjoint union of finite domains? Is this always possible? Please help; any advice will be very appreciated