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Let $(X,\ \tau)$ be a topological (locally compact?) space, $(X,\ \mathcal{F},\ \mu)$ a measure space, $(V,\ \|\cdot\|)$ a Banach space, $1\leq p < \infty$ and $\|\cdot\|_p$ a function defined for a measurable function (with respect to a Borel $\sigma$-algebra on $V$) $f:X\to V$ as

$$\|f\|_p=(\int\limits_X \|f\|^p\mathrm{d}\mu)^\frac{1}{p}.$$

$L^p(X,\ V)$ space is a Banach space of measureable functions $f:X\to V$ such that $\|f\|_p<\infty$, where functions equal almost everywhere are identified, with a norm $\|\cdot\|_p$.

I'm looking for the most general conditions under which a set of compactly-supported continuous functions is dense in $L^p(X,\ V)$. The case of compact $X$ will be the most important for me, so I prefer this constraint to others. Could someone point me to a book or a paper containing such a result?

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There are a few things that one has to consider here

  • Which definition of measurability do you apply here? See http://en.wikipedia.org/wiki/Weakly_measurable_function and http://en.wikipedia.org/wiki/Bochner_measurable_function . In the following, I will assume that we are talking about strong (or Bochner) measurability. This implies that for each $f \in L^p(X;V)$, there is a sequence of simple functions $(f_n)_n$ of the form $$ f_{n}=\sum_{j=1}^{m_{n}}\chi_{A_{j}^{\left(n\right)}}x_{j}^{\left(n\right)} $$ with measurable $A_j^{(n)}$ and $x_j^{(n)} \in V$ satisfying $f_n (x) \to f(x)$ almost everywhere.

    We can assume that the $(A_j^{(n)})_j$ are pairwise disjoint. By replacing $A_j^{(n)}$ by $$ A_{j}^{\left(n\right)}\cap\left\{ x\in X\,\, \middle| \,\, \left|x_{j}^{\left(n\right)}\right|\leq2\left|f\left(x\right)\right|\right\}, $$ we can assume $|f_n (x)| \leq 2 |f(x)|$ (check that still $f_n(x) \to f(x)$ almost everywhere). Also note that the set $\{x \in X \mid 2 |f(x)| \geq |x_j^{(n)}| \}$ is of finite measure for $x_j^{(n)} \neq 0$. We can assume that this is the case for all $j,n$ by dropping the vanishing terms from the sum.

    An application of the dominated convergence theorem shows $f_n \to f$ in $L^p (X;V)$.

We have now reduced the problem to the scalar valued case, because the above shows that it suffices to approximate each $f_n$ and it is then easy to see that it actually suffices to approximate each $\chi_{A_j^{(n)}}$ by a $C_c$ function, because of $$ \left\Vert \sum_{j=1}^{m_{n}}\chi_{A_{j}^{\left(n\right)}}x_{j}^{\left(n\right)}-\sum_{j=1}^{m_{n}}g_{j}^{\left(n\right)}x_{j}^{\left(n\right)}\right\Vert _{p}\leq\sum_{j=1}^{m_{n}}\left\Vert x_{j}^{\left(n\right)}\right\Vert \cdot\left\Vert \chi_{A_{j}^{\left(n\right)}}-g_{j}^{\left(n\right)}\right\Vert _{p}<\varepsilon $$ for suitable $g_j^{(n)} \in C_c$ as long as each scalar valued characteristic function of finite measure(!) can be approximated.

It is easy to see that this condition is also necessary for the denseness of $C_c$ in $L^p(X;V)$, at least for $V \neq \{0\}$.

  • Now, we will put some assumptions on the measure $\mu$ and on the space $X$. The natural condition ensuring what you want is to assume that $X$ is a locally compact Hausdorff space and that $\mu$ is a Radon-measure. This means that $\mu$ is a Borel measure with the following additional properties:

    1. $\mu(K)$ is finite for each $K \subset X$ compact,
    2. $\mu(M) = \inf \{ \mu(U) \mid U \supset M \text{ open} \}$ for all Borel sets $M$.
    3. $\mu(U) = \sup \{\mu(K) \mid K \subset U \text{ compact} \}$ for all open $U$.

    Under these assumptions, Folland, Real Analysis, Proposition 7.9 shows that $C_c$ is dense in the (scalar valued) $L^p$ space. By the considerations above, this suffices. A similar statement is probably also shown in Rudin, Real and Complex Analysis.

Finally, note that Folland, Real Analysis, Proposition 7.8 shows that every Borel measure $\mu$ with $\mu(K) < \infty$ for all compact $K$ is a Radon measure, as long as each open set $U \subset X$ is $\sigma$-compact. This is in particular the case if $X$ is 2nd countable, which is for example the case if $X$ is ($\sigma$-)compact and metrizable.

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  • $\begingroup$ Thank you. May I ask you an additional question? Only now that you mentioned it, I realised that ordinary measureable functions with values in Banach space probably don't form a vector space not to mention a Banach space. But all of my results are actually restricted to a case of values in a finite dimensional normed vector spaces. So my question is this. Do ordinary measurable functions from a measure space into a finite dimensional normed vector space (with a sigma-algebra of Borel sets) form a Banach space under the p-norm? I think it is equivalent to the case of codomain being R^n or C^n. $\endgroup$
    – Tom
    Sep 2 '14 at 10:34
  • $\begingroup$ They form a vector space. $\endgroup$
    – user683848
    May 15 '20 at 8:52

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