I am attempting to clarify Theorem 7.11 in Royden's Real Analysis textbook. The proof concerns the separability of $L^p$ spaces for $1 \leq p < \infty$. They start off by defining $\mathcal S[a,b]$ the step functions on the interval $[a,b]$. Then they define $\mathcal S'$ the step functions with rational coefficients and end points:
$$\mathcal S'[a,b] = \left\{\sum_{j=1}^n \alpha_j \chi_{(a_j,b_j)} : \alpha_j \in \Bbb Q, a_j, b_j \in \Bbb Q \cap [a,b]\right\}$$
$\mathcal S'$ is countable because for each $n$ it has a bijection with a subset of $\Bbb Q^{3n}$. Intuitively, $\mathcal S'$ should be dense in $\mathcal S$ in the $L^p$-sense. However, I am having trouble showing this directly. Below is my attempt.
Let $\phi = \sum_j \alpha_j \chi_{E_j} \in \mathcal S$ be given. Pick $\psi = \sum_j \beta_j \chi_{A_j} \in \mathcal S'$ such that $|\alpha_j – \beta_j| < \varepsilon$ and $\mu(E_j \Delta A_j) < \varepsilon$. Then by two applications of Minkowski's inequality:
\begin{align*}
\lVert \phi – \psi \rVert_p &= \left(\int_a^b|\phi – \psi|^p \mu(dx)\right)^{\frac{1}{p}}\\
&= \left(\int_a^b\left|\sum_{j=1}^n \left|\alpha_j \chi_{E_j} -\beta_j \chi_{A_j}\right| \right|^p \mu(dx)\right)^{\frac{1}{p}}\\
&\leq \sum_{j=1}^n \left(\int_a^b\left|(\alpha_j-\beta_j)\chi_{E_j \cap A_j} + \alpha_j\chi_{E_j-A_j} – \beta_j\chi_{A_j-E_j}\right|^p \mu(dx)\right)^{\frac{1}{p}}\\
&\leq \sum_{j=1}^n\left[ \left(\int_a^b |(\alpha_j-\beta_j)\chi_{E_j \cap A_j}|^p \mu(dx)\right)^{\frac{1}{p}} + \left(\int_a^b \left| \alpha_j\chi_{E_j-A_j} – \beta_j\chi_{A_j-E_j}\right|^p \mu(dx)\right)^{\frac{1}{p}}\right]
\end{align*}
From here I am unsure how to proceed. I am not sure how to bound the second term above by the measure of the set difference. For the first term, I can bound it above by $\mu(E_j \cap A_j) \leq \mu([a,b])$ but I am also not sure if that is the correct approach.
This has been asked before here but I don't think the bounds are correct.
Best Answer
Thank you @Stephen Montgomery-Smith for the suggestion. In accordance with it, let $\varphi = \sum_j\alpha_j \chi_{E_j}$ in $\mathcal S$ be given and pick $\psi = \sum_j \beta_j \chi_{A_i}$ in $\mathcal S'$ and then define $\theta = \sum_j \beta_j \chi_{E_j}$ in $\mathcal S$. Then using the triangle and Minkowski inequalities:
\begin{align*} \lVert \phi - \psi \rVert_p &= \lVert \phi - \theta + \theta - \psi \rVert_p\\ &\leq \lVert \phi - \theta \rVert_p + \lVert \theta - \psi \rVert_p\\ &= \left(\int_a^b\left|\sum_{j=1}^n \left|\alpha_j \chi_{E_j} -\beta_j \chi_{E_j}\right| \right|^p \mu(dx)\right)^{\frac{1}{p}} + \left(\int_a^b\left|\sum_{j=1}^n \left|\beta_j \chi_{E_j} -\beta_j \chi_{A_j}\right| \right|^p \mu(dx)\right)^{\frac{1}{p}}\\ &\leq \sum_{j=1}^n \left(\int_a^b |\alpha_j-\beta_j|^p|\chi_{E_j}|^p \mu(dx)\right)^{\frac{1}{p}} + \sum_{j=1}^n\left(\int_a^b | \beta_j|^p|\chi_{E_j} - \chi_{A_j}|^p \mu(dx)\right)^{\frac{1}{p}}\\ &= \sum_{j=1}^n \left(\int_{E_j} |\alpha_j-\beta_j|^p \mu(dx)\right)^{\frac{1}{p}} + \sum_{j=1}^n \left(\int_{\chi_{E_j \Delta A_j}} | \beta_j|^p \mu(dx)\right)^{\frac{1}{p}}\\ &\leq \sum_{j=1}^n\mu(E_j)^{\frac{1}{p}}|\alpha_j - \beta_j| + \sum_{j=1}^n |\beta_j|\mu(E_j \Delta A_j)^{\frac{1}{p}} \end{align*}
Where we used $|\chi_{E_j} - \chi_{A_j}| = \chi_{E_j \Delta A_j}$ So the appropriate choice of $\beta_j$ is such that: $$|\alpha_j - \beta_j| < \frac{\varepsilon}{2n}\mu(E_j)^{-\frac{1}{p}}$$
And the appropriate choice of $A_j$ is such that: $$\mu(E_j \Delta A_j) < \left(\frac{\varepsilon}{2n |\beta_j|}\right)^p$$
All of which can be found since $\Bbb Q$ is dense in $[a,b]$ and we are only making a finite number of choices from countable sets.