No. There are examples of Borel measures on [itex]\sigma[/itex]-compact (even on compact) spaces which fail to be regular.And I suppose a Borel measure on a [itex]\sigma[/itex]-compact space like the complex plane has to be regular then?
[itex]\mu[/itex] is associated with a bounded linear functional by [itex]\Phi(f) = \int_Xfd\mu[/itex]. Then by the Riesz representation theorem, if X is locally compact & Hausdorff, [itex]\Phi[/itex] is associated with a regular measure [itex]\mu^\prime[/itex] by [itex]\Phi(f) = \int_Xfd\mu^\prime[/itex]. So [itex]\int_Xfd(\mu-\mu^\prime)=0,[/itex] all [itex]f[/itex] in [itex]C_0(X)[/itex].morphism said:Can you explain how you're "regularizing" [itex]\mu[/itex]?
If X is locally compact and Hausdorff can this still happen? example?No. There are examples of Borel measures on [itex]\sigma[/itex]-compact (even on compact) spaces which fail to be regular.
The standard example is X=[0,w] where w is the first uncountable ordinal. This is an exercise in Rudin (last one in chapter 2 if you have the first edition).lark said:If X is locally compact and Hausdorff can this still happen? example?
morphism said:As for your other question, I don't really know what happens in general. I'll think about it some more and let you know if I come up with anything.
Regularizing measures refer to techniques used in statistics and machine learning to prevent overfitting and improve the generalization ability of a model by introducing additional constraints or penalties to the model parameters.
Regularizing measures are important because they help to prevent overfitting, which occurs when a model is overly complex and performs well on the training data but poorly on new data. Regularization helps to balance the trade-off between model complexity and generalization performance, resulting in more reliable and accurate models.
Some common types of regularizing measures include L1 and L2 regularization, dropout, early stopping, and data augmentation. L1 and L2 regularization add penalties to the model parameters to discourage large weights, while dropout randomly drops units from the neural network during training to prevent over-reliance on specific features. Early stopping stops the training process when the model's performance on a validation set stops improving, preventing overfitting. Data augmentation involves generating additional training data by applying transformations to the existing data.
Regularizing measures work by adding constraints or penalties to the model parameters, forcing the model to learn more general patterns in the data instead of memorizing specific examples from the training set. This helps to prevent overfitting and improve the model's ability to generalize to new data.
Regularizing measures should be used when training a model that is prone to overfitting, such as deep neural networks with a large number of parameters. Regularization is especially important when working with small datasets, as these models are more likely to overfit due to the limited amount of data available for training.