Density of Polynomial functions in $L^p$ spaces

Question

Polynomial functions belong to $L^p(\Omega)$ for bounded set $\Omega \subset \mathbb{R^n}.$ Are they dense subsets of $L^p(\Omega)$ for $1 \leq p <\infty$ ? If so, how to prove it? If not, what is the underlying reason?

P.S. : Clearly for $p=\infty$ they are not dense, because the uniform limit of polynomials (continuous function) is continuous.

Answer 1

One can split the argument in three steps:

• Bounded measureable functions are dense in $L^p$: If $f \in L^p$ is non-negative, then $f_n(x):= {min}(f(x),n)$ converges to $f$ in $L^p$ by monotone convergence. Otherwise, write $f = f^+ - f^-$, where $f^+ = max(f,0)$ and $f^-=max(-f,0)$.
• Continuous bounded functions are dense in bounded measureable functions w.r.t. $L^p$-norm. This follows from Lusin's Theorem: If $f$ is Borel and bounded, then for any $\varepsilon>0$ there is some continuous $f_\varepsilon$ such that $\{x : f(x) \neq f_\varepsilon(x)\}$ has measure at most $\varepsilon$ and $||f_\varepsilon||\le ||f||$. Then clearly, $||f_\varepsilon -f ||_p^p \le 2^p ||f||^p \varepsilon$.
• By Stone-Weierstraß theorem, polynomials are dense in continuous functions (since $\Omega$ is bounded)