The Simple Function Approx. Lem. and Theo. for Gen. Meas. Spaces

The Simple Function Approximation Lemma and Theorem for General Measurable Spaces

The Simple Function Approximation Lemma for General Measurable Spaces

Recall from The Simple Function Approximation Lemma that if $f$ is a Lebesgue measurable function defined on a Lebesgue measurable set $E$ and $f$ is bounded on $E$ then for all $\epsilon > 0$ there exists simple functions $\varphi_{\epsilon}$ and $\psi_{\epsilon}$ on $E$ such that:

• 1) $\varphi_{\epsilon} (x) \leq f(x) \leq \psi_{\epsilon}(x)$ on $E$.
• 2) $0 \leq \psi_{\epsilon}(x) - \varphi_{\epsilon}(x) < \epsilon$ on $E$.

This result can be extended to general measurable spaces $(X, \mathcal A)$.

 Lemma 1 (The Simple Function Approximation Lemma for General Measurable Spaces): Let $(X, \mathcal A)$ be a measurable space and let $f$ be a measurable function defined on a measurable set $E$. If $f$ is bounded on $E$ then for all $\epsilon > 0$ there exists simple functions $\varphi_{\epsilon}$ and $\psi_{\epsilon}$ on $E$ such that: 1) $\varphi_{\epsilon}(x) \leq f(x) \leq \psi_{\epsilon}(x)$ on $E$. 2) $0 \leq \psi_{\epsilon}(x) - \varphi_{\epsilon}(x) < \epsilon$ on $E$.

The proof of Lemma 1 is analogous to that of the proof for Lebesgue measurable functions.

The Simple Function Approximation Theorem for General Measurable Spaces

Recall from the The Simple Function Approximation Theorem page that if $f$ is a Lebesgue measurable function defined on a Lebesgue measurable set $E$ then there exists a sequence of simple functions $(\varphi_n(x))_{n=1}^{\infty}$ on $E$ such that:

• 1) $(\varphi_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$.
• 2) $|\varphi_n(x)| \leq |f(x)|$ for all $x \in E$ and for all $n \in \mathbb{N}$.

Furthermore, if $f(x) \geq 0$ then there exists a sequence of pointwise increasing simple functions $(\varphi_n(x))_{n=1}^{\infty}$ on $E$ with the above properties.

We now state an analogous theorem for general measurable spaces.

 Theorem 2 (The Simple Function Approximation Theorem for General Measurable Spaces): Let $(X, \mathcal A)$ be a measurable space and let $f$ be a measurable function defined on a measurable set $E$. Then there exists a sequence of simple functions $(\varphi_n(x))_{n=1}^{\infty}$ on $E$ such that: 1) $(\varphi_n(x))_{n=1}^{\infty}$ converges pointwise to $f(x)$ on $E$. 2) $|\varphi_n(x)| \leq |f(x)|$ for all $x \in E$ and for all $n \in \mathbb{N}$. Furthermore, if $f(x) \geq 0$ on $E$ then there exists a sequence of pointwise increasing simple functions $(\varphi_n(x))_{n=1}^{\infty}$ on $E$ with the above properties.

Once again, the proof is analogous to that of the proof for Lebesgue measurable functions.