  We define direct sums and a corresponding notion of connectedness for graph
limits. Every graph limit has a unique decomposition as a direct sum of
connected components. As is well-known, graph limits may be represented by
symmetric functions on a probability space; there are natural definitions of
direct sums and connectedness for such functions, and there is a perfect
correspondence with the corresponding properties of the graph limit. Similarly,
every graph limit determines an infinite random graph, which is a.s. connected
if and only if the graph limit is connected. There are also characterizations
in terms of the asymptotic size of the largest component in the corresponding
finite random graphs, and of minimal cuts in sequences of graphs converging to
a given limit.
