Is Conditional Value-at-Risk (CVaR) coherent?

When the risk is defined by a discrete random variable, is CVaR a coherent risk measure? I stick to the following definition of CVaR:

$$ CVaR_\alpha(R) = \min_v \quad \left\{ v + \frac{1}{1-\alpha} \mathbb{E}[R-v]^+ \right \}$$

where $R$ is the DISCRETE random variable for the loss and $\alpha$ is the confidence level.

Answer

I found this paper: Conditional value-at-risk for general loss distributions by Rockafellar and Uraysev http://dx.doi.org/10.1016/S0378-4266(02)00271-6

which says CVaR is coherent for general loss distributions, including discrete distributions.

I think that I was confused by other authors who were also confused with the definitions of CVaR. In particular, in the following paper, the author mistakenly stated that Tail Conditional Expectation (TCE) is same as CVaR, and they are not coherent.

http://dx.doi.org/10.1016/S0378-4266(02)00281-9

However, TCE is not same as CVaR in general. If the underlying distribution is continuous, they are same.