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.