Let $\phi_1, \ldots \phi_d: \mathbb{R} \to \mathbb{R}$ be continuos and have compact support. Show that for $$ f := \bigotimes_{i = 1}^{d} \phi_i: \mathbb{R}^d \to \mathbb{R}, \ (x_1, \ldots, x_d) \mapsto \prod_{i = 1}^{d} \phi_i(x_i) $$ we have $$ \int_{\mathbb{R}^d} f(x) dx = \prod_{i = 1}^{d} \int_{\mathbb{R}} \phi_i(x_i) dx_i $$
Proof Since $f$ is continuous and has a compact support as well, we get \begin{align*} \int_{\mathbb{R}^d} f(x) dx & = \int_{\text{supp}(f)} \prod_{i = 1}^{d} \phi_i(x_i) dx \overset{(\ast)}{=} \int_{\text{supp}(\phi_d)} \left( \ldots \left( \int_{\text{supp}(\phi_1)} \prod_{i = 1}^{d} \phi_i(x_i) dx_1 \right) \ldots \right) d x_d \\ & \overset{(\star)}{=} \left( \int_{\text{supp}(\phi_1)} \phi_1(x_1) dx_1 \right) \left( \int_{\text{supp}(\phi_d)} \left( \ldots \left( \int_{\text{supp}(\phi_1)} \prod_{i = 2}^{d} \phi_i(x_i) dx_2 \right) \ldots \right) d x_d \right) \\ & = \ldots = \prod_{i = 1}^{d} \int_{\mathbb{R}} \phi_i(x_i) dx_i \end{align*}
My Question I don't understand the steps $(\ast)$ and especially $(\star)$ . Can we just ''pull out'' a one-dimensional integral since it's just a constant?
The proof is from Otto Forster: Analysis 3 (in German) on page 3. This was a homework assignment and the corrector gave full marks to this answer but suggested using induction instead of the $\ldots$-step. Could someone please show that induction in light of the question above?
Please bear in mind that this is from the very beginning of real analysis III and we have not learned measure theory yet.
The first step follows by a theorem on multiple integrals. Note that we can enclose the domain of integration in a set of the form $[a_1,b_1]\times [a_2,b_2]\times \cdots \times [a_n,b_n]$. In Apostol's book there is a section of 'Evaluation of multiple integral by repeated integration'. The theorem proved here for two variable case extends to any number of variables. If you apply that theorem you will get ($\ast$). About ($\star$): yes, you can pull out $\int\phi_1(x_1)\, dx_1$ because it is a constant. To use induction simply note that when you pull out $\int\phi_1(x_1)\, dx_1$ you are left with a similar integral with $n-1$ variables instead of $n$. By induction hypothesis this integral w.r.t $n-1$ variables can be written as a product of $n-1$ integrals.