  We have analyzed some conditions which are essentially involved in deciding
whether or not a probability distribution is unique (moment-determinate) or
non-unique (moment-indeterminate) by its moments. We suggest new conditions
concerning both absolutely continuous and discrete distributions. By using the
new conditions, which are easily checkable, we either establish new results, or
extend previous ones in both Hamburger case (distributions on the whole real
line) and Stieltjes case (distributions on the positive half-line). Specific
examples illustrate both the results and the relationship between the new
conditions and previously available conditions.
