For a commutative unital ring $R$, and $n\in \mathbb{ N}$, let $\textrm{ SL}_n(R)$ denote the special linear group over $R$, and $\textrm{ E}_n(R)$ the subgroup of elementary matrices. Let ${ \mathcal{ M}}^+$ be the Banach algebra of all complex Borel measures on $[0,+\infty)$ with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is shown that $\textrm{ SL}_n(A)=\textrm{ E}_n(A)$ for Banach subalgebras $A$ of ${ \mathcal{ M}}^+$ that are closed under the operation ${ \mathcal{ M}}^+\owns \mu \mapsto \mu_t$, $t\in [0,1]$, where $\mu_t(E):=\int_E (1-t)^x d\mu(x)$ for $t\in [0,1)$, and Borel subsets $E$ of $[0,+\infty)$, and $\mu_1:=\mu(\{ 0\})\delta$, where $\delta\in { \mathcal{ M}}^+$ is the Dirac measure. Many illustrative examples of such Banach algebras $A$ are given. An example of a Banach subalgebra $A\subset { \mathcal{ M}}^+$, that does not possess the closure property above, but for which $\textrm{ SL}_n(A)=\textrm{ E}_n(A)$ neverthess holds, is also given.
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