Please receive my deep sympathy for the death of your child. Empiema (pus in pleural cavity) is life threatening condition, especially in small kids. Empiema usually occurs as a complication of preexisting lung infection (pneumonia), but it can also occur as part of a generalized blood infection (sepsis). An X-ray image of the chest is the main diagnostic tool for detecting empiema. Auscultation with a stethoscope and percussion (tapping) are not always reliable for diagnosing pleural empiema. Infection spreading in small kids with weakened immunity is very fast. My opinion is that a chest X-ray had to be done at first when doctors gave oxygen to your child. The doctors could have kept it to refer to after further blood analyses. You need an expert opinion from a pathologist or forensic medical expert in order determine if there was any medical/doctor liability.