Loading Bitmaps Efficiently
To achieve that the first thing for me would be to move your logic away from the view in a MVP way.

With that in place you could use a observer/observable pattern to wrap that logic in a different class that would be triggered when that event happen.

The problem with this is when your onMeasure is triggered before the addObserver is ran, this of course would end up with the decodeAndResizeFromResource() not running on that first call. A way to fix that, if it is an issue, could be using CustomPresenter constructor to make the values manually or use some default values and make that first call.

Something like this:

import java.util.Observable;
import java.util.Observer;

public class CustomPresenter extends View implements Observable {

    private Bitmap mBackground;
    private List observers = new ArrayList();

    public CustomPresenter (Context context) {
        super(context);
    }

    public CustomPresenter (Context context, AttributeSet attrs) {
        super(context, attrs);
    }

    public CustomPresenter (Context context, AttributeSet attrs, int defStyleAttr) {
        super(context, attrs, defStyleAttr);
    }

    @Override
    protected void onMeasure(int widthMeasureSpec, int heightMeasureSpec) {
        super.onMeasure(widthMeasureSpec, heightMeasureSpec);
        hasChanged();
        notifyObservers();
    }

    @Override
    protected void onDraw(Canvas canvas) {
        canvas.drawBitmap(mBackground, 0, 0, mPaint);
        super.onDraw(canvas);
    }

    public Map getHugeBitmap(){
        return this.drawable.huge_bitmap;
    }

    public void registerObserver(Observer observer) {
        observers.add(observer);
    }

    public void removeObserver(Observer observer) {
        observers.remove(observer);
    }

    public void notifyObservers() {
        for (Observer ob : observers) {
            ob.update(this);
        }
    }
}


Then the "View"

    public class CustomResizerView implements Observer{
       public void update(Object obj){
           CustomPresenter view = (CustomPresenter) obj;            
           mBackground = decodeAndResizeFromResource(view.getResources(),
                   view.getHugeBitmap, view.getMeasuredWidth(), view.getMeasuredHeight());
       }
}


Then the test

public class Test {
    public static void main(String args[]) {
        CustomPresenter view = new CustomPresenter();
        CustomResizerView resizer = new CustomResizerView();
        view.registerObserver(resizer);
    }
}