# Does my proof show that the sequence of measures defined using mollifiers with shrinking support converges weakly to the Dirac measure?

Following Evan's PDE book, Appendix C4, PP. 629, let's define the function:

$$\eta(x):=C\exp\left(\frac{1}{\|x\|^2 - 1} \right) \forall x \in \mathbb{R}^d$$ when $$\|x\|\leq 1$$ and $$0$$ otherwise, where $$C$$ is a positive constant so that the integral of $$\eta$$ is $$1$$.

Define the mollifier $$\eta_\varepsilon$$ by :$$\eta_\varepsilon(x): \frac{1}{\varepsilon^d}\eta(\frac{x}{\varepsilon})$$.

Next, let's define the measure $$P_\varepsilon$$ by $$dP_\varepsilon(x):=\eta_\varepsilon(x) \, dx$$, where $$dx$$ denotes the Lebesgue measure on $$\mathbb{R}^d$$.

I'm trying to prove (if true?!), that:

$$P_\varepsilon \to \delta_0$$ weakly as $$\varepsilon \to 0$$, where $$\delta_0$$ denotes the Dirac measure at $$0$$. How do we prove this, if possible?

So to start, I want to prove that $$\forall f$$ continuous, bounded on $$\mathbb{R}^d$$, we must have:

\begin{align} & \int_{\mathbb{R}^d} f(x)\, dP_\varepsilon(x) = \int_{\mathbb{R}^d} f(x)\eta_\varepsilon(x) \, dx \\ \to {} & f(0)=\int_{\mathbb{R}^d} f(x) \, d\delta_0(x) \end{align} as $$\varepsilon \to 0$$. To achive this, I've done as follows:

\begin{align} & \int_{\mathbb{R}^d}f(x) \eta_{\varepsilon}(x)\,dx - f(0) \\ = {} & \int_{\mathbb{R}^d}(f(x)-f(0)) \eta_\varepsilon(x) \, dx \\ = {} & \int_{B(0;\varepsilon)}(f(x)-f(0)) \eta_\varepsilon(x) \, dx, \end{align}

as the support of $$\eta_\varepsilon = \bar{B(0;\varepsilon)}$$.

Fix any $$\eta > 0$$. Using the continuity of $$f$$ at $$0$$, we must have a $$\varepsilon > 0$$ so that for all $$x$$ with $$\|x\|\leq \varepsilon$$, $$|f(x)-f(0)|\leq \eta$$. Next, after we've taken the absolute values:

\begin{align} & \left|\int_{B(0;\varepsilon)}(f(x)-f(0)) \eta_\varepsilon(x) \, dx \right| \\ \leq {} & \int_{B(0;\varepsilon)}|f(x)-f(0)| |\eta_\varepsilon(x)| \, dx \\ = {} & \int_{B(0;\varepsilon)}|f(x)-f(0)| \eta_\varepsilon(x) \, dx \\ \leq {} & \eta \int_{B(0;\varepsilon)} \eta_\varepsilon(x)\,dx \\ \leq {} & \eta \int_{\mathbb{R}^d} \eta_\varepsilon(x) \, dx \\ = {} & \eta \end{align}

I think this finishes the proof, but if you could verify if it's correct, then it'd be greatly appreciated :)

• You can't learn all of MathJax or LaTeX in a day, but see my edits for better usage. In particular, putting absolute-value delimiters outside of MathJax while the expression they enclosed was inside MathJax is contraindicated. Jan 27, 2020 at 16:05

For the future, you may want to learn to use the align* environment to format multi-line equations; it looks better than what you have.