Existence of a convergence sequence in weak-star topology

I suppose the underlying measure space is the Lebesgue measure on $\mathbb R^d$.

Here is a sketch of the proof using the Lyapunov convexity theorem.

Let $(v_n)$ be a dense set of $L^1(\mathbb R^d)$. Define $v_0:=\chi_Q$. Then each $v_i$ defines a non-atomic measure via $A \mapsto \int_A v_i\ dx$.

Fix $n$. Then by Lyapunov theorem, there is a set $E_n$ such that $$ \int_{E_n} v_i \ dx =\lambda \int_Q v_i \ dx \quad \forall i =0 \dots n. $$ This creates the sequence $\chi_n := \chi_{E_n}$ of characteristic functions.

Now take $v\in L^1(\mathbb R^d)$ and $\epsilon>0$. Then there is $v_i$ with $\|v-v_i\|_{L^1} \le \epsilon$. Take $n>i$. Then $$ \int_{\mathbb R^d} (\chi_{E_n} - \lambda \chi_Q) v \ dx= \int_{\mathbb R^d} (\chi_{E_n} - \lambda \chi_Q) (v-v_i) \ dx + \int_{\mathbb R^d} (\chi_{E_n} - \lambda \chi_Q) v_i \ dx. $$ By construction of $E_n$ and $n>i$, the second addend is zero. The first is less than $\epsilon$: $$ \left| \int_{\mathbb R^d} (\chi_{E_n} - \lambda \chi_Q) (v-v_i) \ dx \right| \le \|\chi_{E_n} - \lambda \chi_Q\|_{L^\infty} \|v-v_i\|_{L^1} \le 1 \cdot\epsilon. $$ This proves $\chi_{E_n} \rightharpoonup^* \lambda \chi_Q$ in $L^\infty$.