Relevant Passage
I am currently reading Chapter 9 in the third edition of W. Rudin's 'Real and Complex Analysis'. On page 184 Rudin writes:
"On account of 9.7(4) we have \begin{align} (g*h_\lambda)(x)-g(x)&=\int_{-\infty}^\infty[g(x-y)-g(x)]h_\lambda(y)\ dm(y) \\\\ &=\int_{-\infty}^\infty[g(x-y)-g(x)]\lambda^{-1}h_1\left(\frac{y}{\lambda}\right)\ dm(y) \\\\ &=\int_{-\infty}^\infty[g(x-\lambda s)-g(x)]h_1(s)\ dm(s). \end{align} The last integrand is dominated by $2||g||_\infty h_1(s)$ and converges to 0 pointwise for every $s$ as $\lambda\rightarrow0$. Hence (1) follows from the dominated convergence theorem."
Context
The definition given for $h_\lambda(x)$ is: $$ h_\lambda(x)=\int_{-\infty}^\infty e^{itx-\lambda|t|}\ dm(t)\qquad(\lambda>0),\ x\in\mathbb{R}. $$ 9.7(3) and 9.7(4) state $$ h_\lambda(x)=\sqrt{\frac{2}{\pi}}\frac{\lambda}{\lambda^2+x^2},\qquad \int_{-\infty}^\infty h_\lambda(x)\ dm(x)=1. $$ respectively, where $\lambda>0$, $x\in\mathbb{R}$, and $dm(x)$ represents Lebesgue measure on $\mathbb{R}$ divided by $\sqrt{2\pi}$. Also, (1) states $$ \lim_{\lambda\rightarrow0}(g\star h_\lambda)(x)=g(x) $$
The Error[s]
I believe 9.7(4) clearly should be replaced with 9.7(3) in the above passage, and that 9.7(4) should instead be invoked to justify use of the Dominated Convergence Theorem.
It is difficult to find errata for this well written classic, so I thought I might confirm that this is indeed an error on Math SA.