$P(\lvert X - \mu\rvert \geq \sigma)$ as a measure of tailedness I know that one of the standard measures for the "tailedness" of a distribution is kurtosis, i.e. fourth standardized central moment $\frac{\mu_4}{\sigma^4}$. This measure is sort of intuitive to me: as the 4th power "punishes" the outliers much more than 2nd power of the variance does, we would expect a distribution with a heavy tail to have a large kurtosis.
Recently, I thought of using another measure of tailedness--computing the probability content "far away" from the mean, let's call this measure $T$ (for "tail"). Then 
$$T(X) = \mathbb{P}(\lvert X - \mu\rvert\geq\sigma)$$
where $X$ is a random variable whose tailedness we are measuring, with $\mu = \operatorname{E}X$, and $\sigma^2 = \operatorname{Var}(X)$
We could also perhaps use different multiples of $\sigma$ in the definition (which would, however, also bound the value of $T$ to less than $1$ because of Chebyshev's inequality).
Has this measure been used for judging the tailedness of a distribution? What are the properties causing it not to be in (widespread) use? It seems to me like it would be a good way to capture the "Amazon business model phenomenon," where they can exploit the high overall probability content among niche titles:

 A: One attribute of the statistic $P(|X - \mu| \ge \sigma)$ is that it is only giving a single snapshot of "tailedness" that may capture what we want when comparing very specific distributions, but is very misleading for others. As an example, consider a Bernoulli with $p = 0.5$. By this metric, it has a maximally heavy tail, yet most would not consider this a heavy tailed distribution. 
As you noted, this statistic can be adjusted by changing $P(|X - \mu| \ge \sigma)$ to $P(|X - \mu| \ge \lambda \sigma)$. Note that if we make $\lambda = 1 + \epsilon$ instead of $\lambda = 1$, our Bernoulli goes from the heaviest possible tails (by this metric) to the lightest possible tails. This suggests we would need to consider this metric over a range of $\lambda$'s to have meaningful insight into our distribution rather than just a particular value of $\lambda$. 
With that said, evaluating $P(|X - \mu| \ge \lambda \sigma)$ for the values $\lambda = 1, 2, 3,...$ may help provide some interpretable measures of how the tails of a distribution decay, especially given that most analysts are aware that this metric is approximately 0.32, 0.05 and 0.003 for the normal distribution with $\lambda = 1,2,3$. 
A: It is intuitive as you say. But, I don't think it is more informative than the kurtosis because probabilities of such inequalities don't give much hint about the distribution after the boundaries. To be more specific, let's call $Z=\frac{|X-\mu|}{\sigma}$, and calculate $P(Z\geq k)$ as $0.01$. Density of $Z$ after $k$ could be either focused near $k$, or far away from it. In either case, you'd have the same probability. However, $E[Z^4]$ would immensely differ.
A: Heavier-tailed distributions do not necessarily have "more probability in the extreme tails," as is sometimes thought.  See here for a counterexample.https://math.stackexchange.com/a/2510884/472987 
It is the rare, extreme values that characterize heavy tails. The Bernoulli(p) distribution provides a good example:  The smaller the p, the heavier the tail, in the sense that the rare event is more extreme, as measured by number of standard eviations from the mean.
Kurtosis is a good measure of tail heaviness (quantile-based for infinite moments, but for actual data, the ordinary Pearson kurtosis works). 
