I have come across different ways expected shortfall is defined. e.g. $$ES_a(X)=\frac{1}{1-a}\int_a^1VaR_b(X)db$$
and $$ES_a(X)=\frac{1}{a}\int_0^aVaR_b(X)db$$ e.g. on Wikipedia's article.
Are these different?
$VaR$ itself is somewhere defined as simply the quantile and somewhere as negative of quantile.
Could you shed some light on whether it is simply inconsistency of some authors, or is there some deeper reasons?