I am interested in understanding whether a certain property of convolution on $L^p(\mathbb{R})$ spaces can be understood without measure theory. More particularly, consider the vector spaces $C_c(\mathbb{R})$ of compactly supported real-valued continuous functions, and define the norms $$ ||f||_1:=\int_{\mathbb{R}}|f(x)|dx,\qquad ||f||_{\infty}:=\sup_{x\in \mathbb{R}}|f(x)|. $$ and consider the duals (which are necessarily Banach spaces with the operatorial norm) $$ \mathcal{M}(\mathbb{R}):=(C_c(\mathbb{R}),||\cdot||_{L^{\infty}})^*,\qquad L^{\infty}(\mathbb{R}):=(C_c(\mathbb{R}),||\cdot||_{L^{1}})^*. $$ By Holder's Inequality for continuous functions and a soft argument for injectivity we get there are inclusions of $(C_c(\mathbb{R}),||\cdot||_{L^{1}})$ into $\mathcal{M}(\mathbb{R})$ and of $(C_c(\mathbb{R}),||\cdot||_{L^{\infty}})$ into $L^{\infty}(\mathbb{R})$. We then define $$ L^1(\mathbb{R})=\overline{(C_c(\mathbb{R}),||\cdot||_{L^{1}})}\subset \mathcal{M}(\mathbb{R}),\qquad C_0(\mathbb{R})=\overline{(C_c(\mathbb{R}),||\cdot||_{L^{\infty}})}\subset L^{\infty}(\mathbb{R}). $$
Now we consider the convolution of continuous compactly supported functions $$ f*g(x):=\int_{\mathbb{R}} f(x-y)g(y)dx. $$ By Young's Inequality for convolutions we can extend this to continuous maps $$ *\colon L^1\times C_0\to C_0,\qquad *\colon L^1\times L^1\to L^1. $$ In addition, if $\langle f,g\rangle=\int_{\mathbb{R}}f(x)g(x)dx$ and $f^r(x):=f(-x)$ it is easy to compute that for $f,g,h\in C_c(\mathbb{R})$ $$ \langle f*g,h\rangle=\langle f,g^r*h\rangle. $$ Thus, one can define a convolution $$ *\colon \mathcal{M}(\mathbb{R})\times C_0(\mathbb{R})\to L^{\infty}(\mathbb{R}) $$ by specifying $f*g$ to act on $h\in C_0(\mathbb{R})$ by $$ \langle f*g,h\rangle:=\langle f,g^r*h\rangle $$ and this extends the previous definition. I believe that the image of this bilinear map is contained in $C_0(\mathbb{R})$ (I think this is a standard result in measure theory). I'm wondering if anyone knows how prove this using only knowledge of the convolution on $C_c(\mathbb{R})$ and duality-type arguments.