I am trying to write a brief introduction to the Lebesgue integration in $\mathbb{R}^m$ from the general viewpoint. The students do not specialize in this field. So I formulate a theorem without proof.
Theorem. In $\mathbb{R}^m$ there is a $\sigma$-algebra $\Sigma$ and a measure $\mu:\Sigma\to [0,\infty]$ such that
- the space of continuous functions with compact support $C_0(\mathbb{R}^m)$ is dense in $L^1(\mu,\mathbb{R}^m)$;
- the Lebesgue integral matches with the Riemann integral on the space $C_0(\mathbb{R}^m)$;
- if $A\subset B\in\Sigma$ and $\mu(B)=0$ then $A\in\Sigma$.
Almost all goes nice but I can not deduce the equality $\mu(D)=\mu(D+h),\quad D\in\Sigma,\quad h\in\mathbb{R}^m$ from this theorem. I actually can not prove that $D+h$ is measurable whenever $D$ is measurable.
For example let $I_D\in L^1$ then we have $\psi_i\to I_D$ almost everywhere and in $L^1,\quad \psi_i\in C_0(\mathbb{R}^m)$. This implies that $\psi_i\to I_D$ pointwise in $\mathbb{R}^m\smallsetminus Q,\quad\mu(Q)=0$. Then $\widetilde\psi_i(x)=\psi_i(x-h)\to I_{D+h}$ pointwise in $\mathbb{R}^m\smallsetminus (Q+h)$. And I am stuck since the next step requires information on whether $Q+h$ is measurable.
