Market Failure:

The conditions  under which  the welfare  theorems hold constitute 'perfect market'. A situation  in which the theorems fail to hold  is called a market failure.  This  term roughly refers  to conditions under which the free market does not produce optimal welfare,  or, the market mechanism fails to allocate resources  efficiently.  It is  thus a 'failure' compared to a perfect market iconomy.

Definition :  Market  failures are situations in which some of  the assumptions  of  the welfare theorems do not hold and in which, as a consequence, market equilibria cannot be relied on to  yield Pareto optimal outcomes.

Any inefficiencies  that arise  in a market economy must be traceable to a violation of at least one of  the assumptions  of  the theorems.

i) well defined property rights,

ii) universal price quoting of commodities (market completeness), 

iii) price taking  by economic agents (absence of imperfect  competition),

iv) full  information, and

v) convexity of production and consumption sets.

For people  to engage  in mutually beneficial trade,  they must have clear ownership rights and information.  In a perfect market, every good and resource has an owncr and a price, and  the agents have full information available  to  them. Production and consumption technologies  in practice are characterized by  the  absence of indivisibilities  and increasing returns  to scale  - or, more formally, by the absence of non-convexities  in production and consumption sets.