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https://www.planetmath.org/CompactlySupportedContinuousFunctionsAreDenseInLp
Now, it follows easily that any simple function ∑ i = 1 n c i χ A i, where each A i has finite measure, can also be approximated by a compactly supported continuous function. Since this kind of simple functions are dense in L p (X) we see that C c (X) is also dense in L p (X).
https://math.stackexchange.com/questions/465216/space-of-continuous-functions-with-compact-support-dense-in-space-of-continuous
Added: We are trying to show that any continuous function that vanishes at infinity can be approximated closely (within ϵ) by a function with compact support. The range where f is large (greater that ϵ / 2) is some interval (M, N) Our approximating function will agree with f over that range.
https://math.stackexchange.com/questions/242877/compact-support-functions-dense-in-l-1
Namely, to prove that a vector subspace of a Banach space is dense we only need to show that the only continuous linear functional that vanishes on it is the null one. To do this fix $\phi \in L^\infty(\mathbb{R})$ and suppose that $$\tag{1}\int_{-\infty}^\infty \phi(x)f(x)\, dx=0, \quad \forall f\in C_c(\mathbb{R}).$$ We claim that $\phi=0$ almost everywhere.
https://mathoverflow.net/questions/237636/are-compactly-supported-continuous-functions-dense-in-the-continuous-functions-o
Continuous functions on $\mathbb R^d$ such that the support is a compact subset of $\overline{\Omega}$? For "nice" $\Omega$ this would be the space of continuous functions on $\Omega$ vanishing at the boundary. $\endgroup$ – Jochen Wengenroth Apr 29 '16 at 12:50
https://mathproblems123.files.wordpress.com/2011/02/density-1.pdf
Oct 03, 2004 · 1 Approximation by continuous functions In this supplement, we’ll show that continuous functions with compact support are dense in L1 = L1(Rn;m). The support of a complex valued function f on a metric space X is the closure of fx 2 X : f(x) 6= 0g. We’ll denote by Cc(X) the set of all complex valued continuous functions on X with compact support.
https://mathoverflow.net/questions/267710/continuous-functions-dense-in-l-1
If X is a complete doubling metric space equipped with a complete probability measure μ such that all Borel sets are μ -measurable, then Cc(X) --- the continuous functions with compact support --- are dense in L1(μ). Question: What are the weakest conditions under which Cc(X) is dense in L1(μ)...
http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/Chapter%2011-%20Convolutions%20and%20Approximations.pdf
Theorem 11.5 (Continuous Functions are Dense). Let (X,d) be a metric space, τdbe the topology on Xgenerated by dand BX= σ(τd) be the Borel σ—algebra. Suppose µ: BX→[0,∞] is a measure which is σ— finite on τdand let BCf(X) denote the bounded continuous functions on Xsuch that µ(f6=0) <∞.Then
https://www.encyclopediaofmath.org/index.php/Function_of_compact_support
The support of is the closure of the set of points for which is different from zero . Thus one can also say that a function of compact support in is a function defined on such that its support is a closed bounded set located at a distance from the boundary of by a number greater than , …
http://www.math.nthu.edu.tw/~kchen/teaching/5131week3.pdf
dense in Lp(E). These step functions are linear combinations of characteristic functions on some dyadic cubes. This implies that the space of simple functions is also dense in Lp(Rn). In this section we prove that the space of smooth functions with compact supports, and the space of functions with rapidly decreasing derivatives are also dense ...
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