ACR X HAD POSSIBLE LTSS FROM VFR ACFT IN CLASS B AIRSPACE. POSSIBLE SYS ERROR.

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I WAS WORKING THE LCL CTL POS AT LGA. ON FREQ WAS ACR X; A B757 ON A 6 MI FINAL TO RWY 22. ANOTHER TARGET; BEING WORKED BY NEW YORK TRACON; FLEW DIRECTLY UNDERNEATH X; WHICH I WAS WORKING. THE NEW YORK TRACON CTLR RAN HIS TFC; WHICH TURNED OUT TO BE A LIGHT TWIN; RIGHT UNDER MY TFC; IN MY AIRSPACE. HE DID 'POINT OUT' THE TFC; BUT NOT TO ME. OUR RADAR POS CTLR RECEIVED THE 'POINTOUT;' BUT NEGLECTED TO INFORM ME. I NEVER ISSUED A TA BECAUSE AT THE TIME I NOTICED THE TARGET; IT WAS WELL OUTSIDE MY AIRSPACE. I WAS WORKING SEVERAL ACFT AT THE TIME; AND WHEN I WENT BACK TO X TO CLR HIM TO LAND; I NOTICED THE CONFLICTING TFC DIRECTLY UNDER HIM. THE PLT OF X ASKED ABOUT THE TFC THAT WENT UNDER HIM. HE SAW IT ON TCASII AS WELL AS VISUALLY. HE DID RECEIVE AN RA ON TCASII; BUT SINCE HE HAD THE TARGET VISUALLY; HE CONTINUED THE APCH. I ADVISED THE PLT NEW YORK TRACON WAS WORKING THE TFC AND GAVE HIM THE FACILITY PHONE NUMBER TO SPEAK WITH AN AREA SUPVR. SEVERAL FACTORS INVOLVED MADE THIS INCIDENT POTENTIALLY DANGEROUS. 1) NEW YORK TRACON CTLR RUNNING TFC WITHIN SOMEONE ELSE'S AIRSPACE. 2) POINTOUT INFO NEVER GETTING TO THE PERTINENT CTLR. 3) TA INFO NOT ISSUED TO THE PLT. AT THIS TIME; NO FURTHER ACTION IS BEING TAKEN REF THIS INCIDENT.