That is why you have to check. If the P(data|param) is concentrated, then even for uniform priors on the real like you can have P(param|data) is proportional t

That is why you have to check. If the P(data|param) is concentrated, then even for uniform priors on the real like you can have P(param|data) is proportional to a sensible distribution. But you have to check for specific model and data (say P(data|param) = c e^{(data-param)^2}), it isn't enough to verbally work through infinities.