In Chapter III of Loomis' Introduction to Abstract Harmonic Anlaysis, the theory of Daniell integral is discussed and basic concepts are developed. The chapter begins with "we suppose given a vector space $L$ of bounded real-valued functions on a set $S$ and we assume that $L$ is also closed under the lattice operations ...' where the lattice operations means max($\cup$) and min($\cap$).
Now in section 12H, the concept of a monotone family and Baire functions are introduced.
12H. A family of real-valued functions is said to be monotone if it is closed under the operations of taking monotone increasing and monotone decreasing limis. The smallest monotone family including $L$ will be designated by $\mathfrak{B}$ and its members will be called Baire functions.
If $h\leq k$, then any monotone family $\mathfrak{M}$ which contains $(g\cup h)\cap k$ for every $g\in L$ also contains $(f\cup h)\cap k$ for every $f\in\mathfrak{B}$, for the functions $f$ such that $(f\cup h)\cap k\in\mathfrak{M}$ form a monotone family which includes $L$ and therefore includes $\mathfrak{B}$. In particular the smallest monotone family including $L^{+}$ is $\mathfrak{B}^{+}$ (where, for any class of functions $C$, $C^{+}$ is the class of non-negative functions in $C$).
I'm having problems understanding that last paragraph, especially with the last sentence thereof. Now, I assume that $h$ and $k$ are in $L$, and that a simple sequence argument proves the fact that "the functions $f$ such that $(f\cup h)\cap k\in\mathfrak{M}$ form a monotone family".
But where is the fact that $h\leq k$ used? Isn't a similar argument possible without any inequalities concerning $h$ and $k$? And most importantly, I don't see how the last sentence (about $L^{+}$ and $\mathfrak{B}^{+}$) is implied by the foregoing argument. I understand that $\mathfrak{B}^{+}$ is a monotone family containing $L^{+}$, but why is it the smallest such one? I think I'm missing some simple things here... Please enlighten me.