Lp space
Dense subspaces

Throughout this section we assume that 1 ≤ p < ∞. {\displaystyle 1\leq p<\infty.}

Let (S, Σ, μ) {\displaystyle (S,\Sigma,\mu)} be a measure space. An integrable simple function f {\displaystyle f} on S {\displaystyle S} is one of the form f = ∑ j = 1 n a j 1 A j {\displaystyle f=\sum _{j=1}^{n}a_{j}\mathbf {1} _{A_{j}}} where a j {\displaystyle a_{j}} are scalars, A j ∈ Σ {\displaystyle A_{j}\in \Sigma } has finite measure and 1 A j {\displaystyle {\mathbf {1} }_{A_{j}}} is the indicator function of the set A j, {\displaystyle A_{j},} for j = 1, …, n. {\displaystyle j=1,\dots,n.} By construction of the integral, the vector space of integrable simple functions is dense in L p (S, Σ, μ). {\displaystyle L^{p}(S,\Sigma,\mu).}

More can be said when S {\displaystyle S} is a normal topological space and Σ {\displaystyle \Sigma } its Borel 𝜎 – algebra, i.e., the smallest 𝜎 – algebra of subsets of S {\displaystyle S} containing the open sets.

Suppose V ⊆ S {\displaystyle V\subseteq S} is an open set with μ (V) < ∞. {\displaystyle \mu (V)<\infty.} It can be proved that for every Borel set A ∈ Σ {\displaystyle A\in \Sigma } contained in V, {\displaystyle V,} and for every ε > 0, {\displaystyle \varepsilon >0,} there exist a closed set F {\displaystyle F} and an open set U {\displaystyle U} such that F ⊆ A ⊆ U ⊆ V and μ (U) − μ (F) = μ (U ∖ F) < ε {\displaystyle F\subseteq A\subseteq U\subseteq V\quad {\text{and}}\quad \mu (U)-\mu (F)=\mu (U\setminus F)<\varepsilon }

It follows that there exists a continuous Urysohn function 0 ≤ φ ≤ 1 {\displaystyle 0\leq \varphi \leq 1} on S {\displaystyle S} that is 1 {\displaystyle 1} on F {\displaystyle F} and 0 {\displaystyle 0} on S ∖ U, {\displaystyle S\setminus U,} with ∫ S | 1 A − φ | d μ < ε. {\displaystyle \int _{S}|\mathbf {1} _{A}-\varphi |\,\mathrm {d} \mu <\varepsilon \,.}

If S {\displaystyle S} can be covered by an increasing sequence (V n) {\displaystyle (V_{n})} of open sets that have finite measure, then the space of p {\displaystyle p} – integrable continuous functions is dense in L p (S, Σ, μ). {\displaystyle L^{p}(S,\Sigma,\mu).} More precisely, one can use bounded continuous functions that vanish outside one of the open sets V n. {\displaystyle V_{n}.}

This applies in particular when S = R d {\displaystyle S=\mathbb {R} ^{d}} and when μ {\displaystyle \mu } is the Lebesgue measure. The space of continuous and compactly supported functions is dense in L p (R d). {\displaystyle L^{p}(\mathbb {R} ^{d}).} Similarly, the space of integrable step functions is dense in L p (R d); {\displaystyle L^{p}(\mathbb {R} ^{d});} this space is the linear span of indicator functions of bounded intervals when d = 1, {\displaystyle d=1,} of bounded rectangles when d = 2 {\displaystyle d=2} and more generally of products of bounded intervals.

Several properties of general functions in L p (R d) {\displaystyle L^{p}(\mathbb {R} ^{d})} are first proved for continuous and compactly supported functions (sometimes for step functions), then extended by density to all functions. For example, it is proved this way that translations are continuous on L p (R d), {\displaystyle L^{p}(\mathbb {R} ^{d}),} in the following sense: ∀ f ∈ L p (R d): ‖ τ t f − f ‖ p → 0, as R d ∋ t → 0, {\displaystyle \forall f\in L^{p}\left(\mathbb {R} ^{d}\right):\quad \left\|\tau _{t}f-f\right\|_{p}\to 0,\quad {\text{as }}\mathbb {R} ^{d}\ni t\to 0,} where (τ t f) (x) = f (x − t). {\displaystyle (\tau _{t}f)(x)=f(x-t).}
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