This is Exercise $51$ in Tao’s blog: https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/. Define a good kernel to be a measurable function ${P: {\mathbb R}^d \rightarrow {\mathbb R}^+}$ which is non-negative, radial (which means that there is a function ${\tilde P: [0,+\infty) \rightarrow {\mathbb R}^+}$ such that ${P(x) = \tilde P(|x|)})$, radially non-increasing (so that ${\tilde P}$ is a non-increasing function), and has total mass ${\int_{{\mathbb R}^d} P(x)\ dx}$ equal to ${1}$. The functions ${P_t(x) := \frac{1}{t^d} P(\frac{x}{t})}$ for ${t>0}$ are then said to be a good family of approximations to the identity.
Show that the heat kernels ${P_t(x) := \frac{1}{(4\pi t^2)^{d/2}} e^{-|x|^2/4t^2}}$ and Poisson kernels ${P_t(x) := c_d \frac{t}{(t^2+|x|^2)^{(d+1)/2}}}$ are good families of approximations to the identity, if the constant ${c_d > 0}$ is chosen correctly.
Question 1: For the heat kernel, $P(x) = \frac{1}{(4\pi)^{d/2}} e^{-|x|^2/4}$. How do we show that the total mass is $1$?
Question 2: How to show that the Poissson kernels are good families of approximations to the identity, depending on $c_d$?
For some further hint by Tao himself：For the heat kernel, this is a gaussian integral and can be computed by a number of techniques (for instance a reduction to the famous identity $\int_{-\infty}^\infty e^{-\pi x^2}\ dx = 1).$ For the Poisson kernel, one can rescale $t=1$,and manipulate the integral by a number of techniques (e.g., polar coordinates and contour shifting, or using the identity $\frac{\Gamma(s)}{a^s} = \int_0^\infty e^{-at} t^{s-1}\ dt$ to express the Poisson kernel in terms of gaussians); alternatively, one can simply just declare $c_d$ by fiat to normalize the mass of the Poisson kernel.
For (2), one can obtain pointwise upper and lower bounds for $P$ in terms of the horizontal wedding cake function mentioned, as well as a dilate of this function (replacing $x$ by $x/2$ or $2x$).
For the heat kernel notice that if you consider $(X_{1},\dots,X_{d})$ a vector of i.i.d gaussian random variables with mean $0$ and variance $\sqrt 2$ , the heat kernel is precisely the pdf of the vector and as such, its mass must be 1.
For the poisson kernel, I assume your confusion is on how to choose the appropriate constant.
Note that if the mass of the function $g(x)=\frac{1}{(1+|x|)^{(d+1)/2}}$ is finite then the value of $c_{d}$ is forced, it's the only number that makes it so $c_{d}g(x)=P(x)$ has mass 1.