The monotone class theorem is a result in measure theory which allows statements about particularly simple classes of functions to be generalized to arbitrary measurable and bounded functions.

That $\mathcal{H}$ is a vector space just means that it is closed under taking linear combinations, so $\lambda f+\mu g\in\mathcal{H}$ whenever $f,g\in\mathcal{H}$ and $\lambda,\mu\in\mathbb{R}$.

As an example application, consider Fubini’s theorem, which states that for any two finite measure spaces $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ then we may commute the order of integration

Here, $f\colon X\times Y\rightarrow\mathbb{R}$ is a bounded and $\mathcal{A}\otimes\mathcal{B}$-measurable function. The space $\mathcal{H}$ of functions for which this identity holds is easily shown to be linearly closed and, by the monotone convergence theorem, is closed under taking monotone limits of functions. Furthermore, the $\pi$-system of sets of the form $A\times B$ for $A\in\mathcal{A}$, $B\in\mathcal{B}$ generates the $\sigma$-algebra $\mathcal{A}\otimes\mathcal{B}$ and

So, $1_{{A\times B}}\in\mathcal{H}$ and the monotone class theorem allows us to conclude that all real valued and bounded $\mathcal{A}\otimes\mathcal{B}$-measurable functions are in $\mathcal{H}$, and equation (1) is satisfied.

An alternative, more general form of the monotone class theorem is as follows. The version of the monotone class theorem given above follows from this by letting $\mathcal{K}$ be the characteristic functions $1_{A}$ for $A\in\mathcal{S}$.

Saying that $\mathcal{A}$ is the $\sigma$-algebra generated by $\mathcal{K}$ means that it is the smallest $\sigma$-algebra containing the sets $f^{{-1}}(B)$ for $f\in\mathcal{K}$ and Borel subset $B$ of $\mathbb{R}$.

For example, letting $\mathcal{K}$ be as in theorem 2 and $\mu,\nu$ be finite measures on $(X,\mathcal{A})$ such that $\mu(X)=\nu(X)$ and $\int f\,d\mu=\int f\,d\nu$ for all $f\in\mathcal{K}$, then $\int f\,d\mu=\int f\,d\nu$ for all bounded and measurable real valued functions $f$. Therefore, $\mu=\nu$. In particular, a finite measure $\mu$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is uniquely determined by its characteristic function $\chi(x)\equiv\int e^{{ixy}}\,d\mu(y)$ for $x\in\mathbb{R}$. Similarly, a finite measure $\mu$ on a bounded interval is uniquely determined by the integrals $\int x^{{n}}\,d\mu(x)$ for $n\in\mathbb{Z}_{+}$.