Is this a Sobolev space?

Question

Let $(\Omega,\mathcal F, P)$ be a measure space with $\Omega\subseteq\mathbb R^n$. Furthermore, let $\mathcal C^p(\Omega,\mathbb R)$ the space of all $p$-times continuously differentiable functions $\Omega\rightarrow\mathbb R$. Is the space $$\left\{f\in\mathcal C^p(\Omega,\mathbb R) : \int_\Omega \big\vert\mathrm D^\alpha f(\omega)\big\vert\,\mathrm d\mu<\infty~~\text{for all $\alpha\in(\mathbb N_0)^p$ with $\vert\alpha\vert\leq p$} \right\}$$ a Sobolev space $\mathcal W^{\ell, m}$ for a particular choice of $m,\ell$? That is, the space of all continuously differentiable functions with integrable partial derivatives. Here $\mathrm D^\alpha$ denotes the (strong) derivative operator $\frac{\partial^{\vert\alpha\vert}}{\partial^{\alpha_1}\partial^{\alpha_2}\cdots\partial^{\alpha_p}}$.