Given two discrete distributions $P$ and $Q$ with the same support $x\_1,\cdots,x\_n$. Assume $K \in L^1(\mathbb{R})$ is a nonnegative function with $\int\_\mathbb{R} K(x)dx = 1$, and let $K\_h(x) = \frac{1}{h}K(\frac{x}{h})$. 


I am wondering whether the following result holds:
$$\left| \int\_\mathbb{R} |P\* K\_h(x) - Q \* K\_h(x)|dx - \sum\_i|P\_i - Q\_i| \right| \rightarrow 0, \quad as \, h\rightarrow 0 \, ?$$
where $P\*K\_h = \sum\_i P\_iK\_h(x-x\_i)$ is the convolution between $P$ and $K\_h$. In other words, I would like $\left\Vert P\*K\_h - Q\*K\_h\right\Vert\_1$ to be close to $\left\Vert P-Q\right\Vert\_1$ when $h$ is small.


Actually we have that 
$$\int\_\mathbb{R} |P\* K\_h(x) - Q \* K\_h(x)|dx \leq \sum\_i|P\_i - Q\_i| \int K\_h(x-x\_i)dx = \sum\_i|P\_i - Q\_i|,$$
but I do not know what is the case for the opposite direction.


I am also wondering will the same statement still hold when the $\ell\_1$ distance is replaced with other distances, like Jensen–Shannon divergence.


Any hint would be appreciated.



L1 distance after Convolution
This is definitely true, and the point is that for any two distinct $x\neq y\in \mathbb R$ the measures $\delta\_x\*K\_h$ and $\delta\_y\* K\_h$ are asymptotically singular as $h\to 0$, i.e.,
$$
\| \delta\_x\*K\_h - \delta\_y\*K\_h \| \to 2 \qquad \text{as}\quad h\to 0 \;.
$$


