202405241514
Status: #idea
Tags: Measure Theory
State: #nascient

Monotone Classes

They are a concept strongly connected to $σ$ -algebras. In fact, a field (algebra ) $B$ is a $σ - a l g e b r a$ if and only if it is also a monotone class. A class is another word for algebras.

SO what the heck is a monotone class? A monotone class is a set that is closed under monotone limits of increasing and decreasing sets.

We must note that for increasing sets their limit is given by:

⋃_{n} A_{n}

while for increasing their limit is given by:

⋂_{n} A_{n}

In other words, a class is a monotone class if the union of a sequence of increasing sets (sets such that $A_{i} \subseteq A_{j}$ for all $i <= j$ ) is in the class AND if the intersection of a sequence of decreasing sets (sets such that $A_{i} \supseteq A_{j}$ for all $i <= j$ ) are in the class.

Monotone Classes

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