I am studying Stanislaw Lojasiewicz book – “An introduction to the Theory of Real Functions” and I do not uderstand few things. I hope you’ll help me. Here is what is written:

G is an open set and $Gamma(G)$ is defined as a class of all continous, non-negative functions on compact space such that $varphileq 1$ and cl${x:varphi(x)neq 0}subset G$,

$lambda(G)=sup_{Gamma(G)}I(varphi).$

Now let $G_n$ be a sequence of open sets. Let $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$. Then there exists $varphiinGamma(bigcuplimits_{i=1}^{infty} G_{i})$ such that $L<I(varphi).$

That’s first thing. Why can we define such constant? How can we know it exist? And why existing of constant $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$ implies fact that $L<I(varphi).$

I will be really thankful for any advices.

*Łojasiewicz, Stanisław*, An introduction to the theory of real functions. Transl. from the Polish by G. H. Lawden, ed. by A. V. Ferreira, Wiley-Interscience Publication. Chichester (UK) etc.: Wiley. ix, 230 p. textsterling 24.95 (1988). ZBL0653.26001.