The paper discusses the coherence of Expected Shortfall (ES) as a risk measure, which is proposed as a remedy for the deficiencies of Value-at-Risk (VaR). VaR is criticized for not being sub-additive, meaning that the risk of a portfolio can be larger than the sum of the individual risks of its components. To address this issue, the concept of coherent risk measures was introduced, with the worst conditional expectation (WCE) being a coherent example. However, WCE and other related measures like tail conditional expectation (TCE) and conditional value-at-risk (CVaR) can fail to be coherent when applied to discontinuous loss distributions. The authors compare several definitions of ES and find that one definition is robust and yields a coherent risk measure regardless of the underlying distributions. This definition is shown to be effective even in cases where traditional VaR estimators fail. The paper also provides characterizations of ES, including its representation as the integral of all quantiles below a specified level, as the limit in a tail strong law of large numbers, and as the maximum of WCEs when the underlying probability space varies. Key properties of ES are discussed, such as its coherence, continuity, and monotonicity in the confidence level. The paper highlights that ES is continuous with respect to the confidence level, making it insensitive to small changes in the confidence level. Additionally, it is shown that ES coincides with CVaR under certain conditions, and it can be represented as the expectation of a suitably modified tail distribution. The authors also present inequalities and counter-examples to illustrate the relationships between ES, TCE, and WCE. An example demonstrates that none of these measures can define a sub-additive risk measure in general. Finally, the paper concludes by showing that ES can be represented in terms of WCEs, making it insensitive to the underlying probability space when the latter is "small" in a specific sense.The paper discusses the coherence of Expected Shortfall (ES) as a risk measure, which is proposed as a remedy for the deficiencies of Value-at-Risk (VaR). VaR is criticized for not being sub-additive, meaning that the risk of a portfolio can be larger than the sum of the individual risks of its components. To address this issue, the concept of coherent risk measures was introduced, with the worst conditional expectation (WCE) being a coherent example. However, WCE and other related measures like tail conditional expectation (TCE) and conditional value-at-risk (CVaR) can fail to be coherent when applied to discontinuous loss distributions. The authors compare several definitions of ES and find that one definition is robust and yields a coherent risk measure regardless of the underlying distributions. This definition is shown to be effective even in cases where traditional VaR estimators fail. The paper also provides characterizations of ES, including its representation as the integral of all quantiles below a specified level, as the limit in a tail strong law of large numbers, and as the maximum of WCEs when the underlying probability space varies. Key properties of ES are discussed, such as its coherence, continuity, and monotonicity in the confidence level. The paper highlights that ES is continuous with respect to the confidence level, making it insensitive to small changes in the confidence level. Additionally, it is shown that ES coincides with CVaR under certain conditions, and it can be represented as the expectation of a suitably modified tail distribution. The authors also present inequalities and counter-examples to illustrate the relationships between ES, TCE, and WCE. An example demonstrates that none of these measures can define a sub-additive risk measure in general. Finally, the paper concludes by showing that ES can be represented in terms of WCEs, making it insensitive to the underlying probability space when the latter is "small" in a specific sense.